UNIT 3
Proportionality:
Ratios and
Rates
MODULE
MODULE
7
Representing Ratios
and Rates
6.4.B, 6.4.C, 6.4.D,
6.4.E
8
Applying Ratios
MODULE
MODULE
and Rates
6.4.A, 6.4.H, 6.5.A
9
Percents
MODULE
MODULE
CAREERS IN MATH
Residential Builder A residential
builder, also called a homebuilder, specializes
in the construction of residences that range
from single-family custom homes to buildings
that contain multiple housing units, such as
apartments and condominiums. Residential
builders use math in numerous ways, such
as blueprint reading, measuring and scaling,
using ratios and rates to calculate the amounts
of different building materials needed, and
estimating costs for jobs.
Choice/Getty Images
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Alexander Hafemann/Photographer’s
6.4.E, 6.4.F, 6.4.G,
6.5.B, 6.5.C, 6.5.G
Unit 3 Performance Task
At the end of the unit,
check out how residential
builders use math.
If you are interested in a career as a residential
builder, you should study these mathematical
subjects:
• Algebra
• Geometry
• Business Math
• Technical Math
Research other careers that require using ratios
and rates, and measuring and scaling.
Unit 3
175
Preview
UNIT 3
Vocabulary
Use the puzzle to preview key vocabulary from this unit. Unscramble the circled
letters within found words to answer the riddle at the bottom of the page.
1
2
3
4
5
6
Down
1. A rate that describes how much smaller or
larger the scale drawing is than the real object.
(Lesson 8-3)
1. Drawing that uses a scale to make an object
proportionally smaller or larger than the real
object. (Lesson 8-3)
4. A multiplicative comparison of two quantities
expressed with the same units.
(Lesson 7-1)
2. A rate in which the second quantity is one
unit. (Lesson 7-2)
5. A comparison by division of two quantities
that have different units. (Lesson 7-2)
3. A fraction that compares two equivalent
measurements. (Lesson 8-4)
6. An equation that states two ratios are
equivalent. (Lesson 8-3)
Q:
Why was the draftsman excited that the raffle prize was a
weighing device?
A: It was a
176
Vocabulary Preview
–
!
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Across
Representing
Ratios and Rates
?
MODULE
7
LESSON 7.1
ESSENTIAL QUESTION
Ratios
How can you use ratios
and rates to solve
real-world problems?
6.4.C, 6.4.E
LESSON 7.2
Rates
6.4.D
LESSON 7.3
Using Ratios and
Rates to Solve
Problems
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Palmer / Alamy
6.4.B
Real-World Video
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Scientists studying sand structures determined that
the perfect sand and water mixture is equal to 1
bucket of water for every 100 buckets of sand. This
1
recipe can be written as the ratio ___
.
100
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Animated Math
Personal Math Trainer
Go digital with your
write-in student
edition, accessible on
any device.
Scan with your smart
phone to jump directly
to the online edition,
video tutor, and more.
Interactively explore
key concepts to see
how math works.
Get immediate
feedback and help as
you work through
practice sets.
177
Are YOU Ready?
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Complete these exercises to review skills you will need
for this chapter.
Simplify Fractions
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15
·
Simplify __
24
EXAMPLE
15: 1, 3 , 5, 15
24: 1, 2, 3 , 4, 6, 8, 12, 24
15
÷3
_____
= _58
24 ÷ 3
Online
Assessment and
Intervention
List all the factors of the numerator
and denominator.
Circle the greatest common factor (GCF).
Divide the numerator and denominator
by the GCF.
Write each fraction in simplest form.
1. _69
4
2. __
10
15
3. __
20
20
4. __
24
16
5. __
56
45
6. __
72
18
7. __
60
32
8. __
72
Write Equivalent Fractions
EXAMPLE
6
×2
_
= 6____
8
8×2
12
= __
16
÷2
6
_
= 6____
8
8÷2
Multiply the numerator and denominator by the same
number to find an equivalent fraction.
Divide the numerator and denominator by the same
number to find an equivalent fraction.
= _34
12 = _____
9. ___
15
5
15 = ______
13. ___
40
8
178
Unit 3
5 = ______
10. __
6
30
16 = ______
4
11. ___
24
3 = _____
21
12. __
9
18 = ______
14. ___
30
10
48 = ______
12
15. ___
64
18
2 = ______
16. __
7
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Write the equivalent fraction.
Reading Start-Up
Visualize Vocabulary
Use the ✔ words to complete the chart. Choose the review
words that describe multiplication and division.
Understanding Multiplication and Division
Symbol
Operation
Term for the answer
×
÷
Vocabulary
Review Words
colon (dos puntos)
denominator
(denominador)
✔ divide (dividir)
fraction bar (barra de
fracciones)
✔ multiply (multiplicar)
numerator (numerador)
✔ product (producto)
quantity (cantidad)
✔ quotient (cociente)
term (término)
Preview Words
Understand Vocabulary
Match the term on the left to the definition on the right.
1. rate
A. Rate in which the second
quantity is one unit.
2. ratio
B. Multiplicative comparison of two
quantities expressed with the same
units.
3. unit rate
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4. equivalent
ratios
equivalent ratios
(razones equivalentes)
rate (tasa)
ratio (razón)
unit rate (tasa unitaria)
C. Ratios that name the same comparison.
D. Comparison by division of two
quantities that have different units.
Active Reading
Two-Panel Flip Chart Create a two-panel flip
chart, to help you understand the concepts in
this module. Label one flap “Ratios” and the
other flap “Rates.” As you study each lesson,
write important ideas under the appropriate
flap. Include information about unit rates
and any sample equations that will help you
remember the concepts when you look back
at your notes.
Module 7
179
MODULE 7
Unpacking the TEKS
Understanding the TEKS and the vocabulary terms in the TEKS
will help you know exactly what you are expected to learn in
this module.
6.4.B
Apply qualitative and
quantitative reasoning to solve
prediction and comparison of
real-world problems involving
ratios and rates.
Key Vocabulary
rate (tasa)
A comparison by division of
two quantities measured in
different units.
What It Means to You
You will solve real-world problems involving rates.
UNPACKING EXAMPLE 6.4.B
A group of 10 friends is in line to see a movie. The table shows how
much different groups will pay in all. Predict how much the group
of 10 will pay.
Number in group
3
5
6
12
Amount paid ($)
15
25
30
60
The rates are all the same.
3
__
= _15
15
6
__
= _15
30
5
__
= _15
25
12 _
__
= 15
60
Find which number in the group is equal to _15 .
10 _
__
= 15
?
→
10
÷ 10
______
= _15
50 ÷ 10
→
10 _
__
= 15
50
6.4.D
Give examples of rates as the
comparison by division of two
quantities having different
attributes, including rates as
quotients.
Key Vocabulary
unit rate (tasa unitaria)
A rate in which the second
quantity in the comparison is
one unit.
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the
unpacked.
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180
Unit 3
What It Means to You
You will solve problems involving unit rates by division.
UNPACKING EXAMPLE 6.4.D
A 2-liter bottle of soda costs $2.02. A 3-liter bottle of the
same soda costs $2.79. Which is the better deal?
2-liter bottle
3-liter bottle
$2.02
_____
2 liters
$2.02 ÷ 2
________
2 liters ÷ 2
$1.01
_____
1 liter
$2.79
_____
3 liters
$2.79 ÷ 3
________
3 liters ÷ 3
$0.93
_____
1 liter
The 3-liter bottle is the better deal.
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A group of 10 will pay $50.
LESSON
7.1 Ratios
?
Proportionality—
6.4.C Give examples
of ratios as multiplicative
comparisons of two quantities
describing the same attribute.
Also 6.4.E
ESSENTIAL QUESTION
How do you use ratios to compare two quantities?
EXPLORE ACTIVITY
6.4.E
Representing Ratios with Models
A ratio is a multiplicative comparison of two quantities expressed with
the same units. The figure shows a ratio of 4 blue squares to 1 red
square, or 4 to 1.
A bracelet has 3 star-shaped beads for every 1 moon-shaped bead.
A Write the ratio of star beads to moon beads.
B If the bracelet has 2 moon beads, how many star beads
does it have?
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C If the bracelet has 9 star beads, how many moon beads does it have?
How do you know?
Reflect
1.
Make a Prediction Write a rule that you can use to find the number
of star beads when you know the number of moon beads.
2.
Make a Prediction Write a rule that you can use to find the number
of moon beads when you know the number of star beads.
Lesson 7.1
181
Writing Ratios
The numbers in a ratio are called terms. A ratio can be written in several
different ways.
Math On the Spot
5 dogs to 3 cats
5 to 3
5:3
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5
_
3
Ratios can be written to compare a part to a part, a part to the whole, or the
whole to a part.
EXAMPLE 1
6.4.C
A Write the ratio of comedies to cartoons
in three different ways.
Math Talk
Mathematical Processes
What does it mean
when the terms in a ratio
are equal?
Part to part
8:2
8
_
2
8 comedies to 2 cartoons
B Write the ratio of dramas to total videos
in three different ways.
Sam’s Video Collection
Comedies
8
Dramas
3
Cartoons
2
Science Fiction
1
Part to whole
3 to 14
3 : 14
3
__
14
The total number of videos
is 8 + 3 + 2 + 1 = 14.
3.
Analyze Relationships Describe the relationship between the drama
videos and the science fiction videos.
4.
Analyze Relationships The ratio of floor seats to balcony seats in a
theater is 20 : 1. Does this theater have more floor seats or more balcony
seats? How do you know?
YOUR TURN
Write each ratio in three different ways.
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Unit 3
5.
bagel chips to peanuts
6.
total party mix to pretzels
7.
cheese crackers to peanuts
Party Mix
Makes 8 cups
4 cups pretzels
2 cups bagel chips
1 cup cheese crackers
1 cup peanuts
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Reflect
Equivalent Ratios
Equivalent ratios are ratios that name the same comparison. You can find
equivalent ratios by using a table or by multiplying or dividing both terms
of a ratio by the same number.
×2
÷4
2
_
7
8
__
24
4
__
14
Math On the Spot
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2
_
6
÷4
×2
EXAMPL 2
EXAMPLE
6.4.C
A punch recipe makes 5 cups of punch by mixing 3 cups of cranberry juice
with 2 cups of apple juice. How much cranberry juice and apple juice do
you need to make four times the original recipe?
Method 1
STEP 1
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STEP 2
Use a table.
Make a table comparing the amount
of cranberry juice and apple juice
needed to make two times, three
times, four times, and five times the
original recipe.
Multiply both terms of
the original ratio by the
same number to find an
equivalent ratio.
2×3
↓
3×3
↓
4×3
↓
5×3
↓
Cranberry Juice
3
6
9
12
15
Apple Juice
2
4
6
8
10
↑
2×2
↑
3×2
↑
4×2
↑
5×2
Write the original ratio and the ratio that shows
the amount of cranberry juice and apple juice
needed to make four times the original recipe.
Math Talk
3
12
_
= __
2
8
Method 2
STEP 1
Mathematical Processes
Multiply both terms of the ratio by the same number.
Write the original ratio in fraction form.
3
_
2
STEP 2
My Notes
The ratio of apple juice to
grape juice in a recipe is 8 cups
to 10 cups. How can you find
the amount of each juice
needed if the recipe
is cut in half?
Multiply the numerator and denominator by the same number.
To make four times the original recipe, multiply by 4.
×4
3
_
2
12
__
8
×4
To make four times the original recipe, you will need 12 cups of
cranberry juice and 8 cups of apple juice.
Lesson 7.1
183
YOUR TURN
Personal
Math Trainer
Online Assessment
and Intervention
Find three ratios equivalent to the given ratio.
8
__
10
8.
9.
5
_
2
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Guided Practice
The number of dogs compared to the number of cats owned by the
residents of an apartment complex is represented by the model shown.
(Explore Activity)
1. Write a ratio that compares the number of dogs to
the number of cats.
2. If there are 15 cats in the apartment complex, how
many dogs are there?
15 ÷
=
dogs
3. How many cats are there if there are 5 dogs in the
apartment complex?
=
cats
The contents of Dana’s box of muffins is shown. Write each ratio in
three different ways. (Example 1)
4. Banana nut muffins to chocolate chip muffins
5. Bran muffins to total muffins
Write three equivalent ratios for the given ratio. (Example 2)
10
6. __
12
?
?
14
7. __
2
8. _47
ESSENTIAL QUESTION CHECK-IN
9. Use an example to describe the multiplicative relationship between two
equivalent ratios.
184
Unit 3
Dana’s Dozen Muffins
6 chocolate chip
3 bran
2 banana nut
1 blueberry
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Date
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10. Draw a model to represent the ratio 1 to 3. Describe how to use the model
to find an equivalent ratio.
20
11. The ratio of boys to girls on the bus is __
15 . Find three ratios equivalent to
the described ratio.
12. In each bouquet of flowers, there
are 4 roses and 6 white carnations.
Complete the table to find how many
roses and carnations there are in
4 bouquets of flowers.
Roses
4
Carnations
6
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13. Ed is using the recipe shown to make fruit salad. He wants to use
30 diced strawberries in his fruit salad. How many bananas, apples,
and pears should Ed use in his fruit salad?
14. A collector has 120 movie posters and 100 band posters. She wants
to sell 24 movie posters but still have her poster collection maintain
the same ratio of 120 : 100. If she sells 24 movie posters, how many band
posters should she sell? Explain.
Fruit Salad Recipe
4 bananas, diced
3 apples, diced
6 pears, diced
10 strawberries, diced
15. Bob needs to mix 2 cups of liquid lemonade concentrate with 3.5 cups of
water to make lemonade. Bob has 6 cups of lemonade concentrate. How
much lemonade can he make?
16. Multistep The ratio of North American butterflies to South American
butterflies at a butterfly park is 5 : 3. The ratio of South American butterflies
to European butterflies is 3 : 2. There are 30 North American butterflies at
the butterfly park.
a. How many South American butterflies are there?
b. How many European butterflies are there?
Lesson 7.1
185
17. Sinea and Ren are going to the carnival next week. The table shows the
amount that each person spent on snacks, games, and souvenirs the last
time they went to the carnival.
Snacks
Games
Souvenirs
Sinea
$5
$8
$12
Ren
$10
$8
$20
a. Sinea wants to spend money using the same ratios as on her last trip
to the carnival. If she spends $26 on games, how much will she spend
on souvenirs?
b. Ren wants to spend money using the same ratios as on his last trip to
the carnival. If he spends $5 on souvenirs, how much will he spend on
snacks?
c. What If? Suppose Sinea and Ren each spend $40 on snacks, and
each person spends money using the same ratios as on their last trip.
Who spends more on souvenirs? Explain.
FOCUS ON HIGHER ORDER THINKING
Work Area
19. Analyze Relationships How is the process of finding equivalent ratios
like the process of finding equivalent fractions?
20. Explain the Error Tina says that 6 : 8 is equivalent to 36 : 64. What did Tina
do wrong?
186
Unit 3
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18. Communicate Mathematical Ideas Explain why the ratio 2 to 5 is
different from the ratio 5 to 2 if both represent the ratio of cats to dogs.
LESSON
7.2 Rates
?
Proportionality—
6.4.D Give examples of
rates as the comparison by
division of two quantities
having different attributes,
including rates as quotients.
ESSENTIAL QUESTION
How do you use rates to compare quantities?
6.4.D
EXPLORE ACTIVITY
Using Rates to Compare Prices
A rate is a comparison by division of two quantities that have different units.
Chris drove 107 miles in two hours. You are comparing miles and hours.
Math Talk
Mathematical Processes
107 miles
.
The rate is _______
2 hours
Shana is at the grocery store comparing two brands of juice. Brand A costs
$3.84 for a 16-ounce bottle. Brand B costs $4.50 for a 25-ounce bottle.
Ryan drove more hours
than Chris at the same rate
of speed. Who drove the
most miles? Explain.
To compare the costs, Shana must compare prices for equal amounts of juice.
How can she do this?
A Complete the tables.
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Brand A
Brand B
Ounces
Price ($)
÷2
16
3.84
÷2
8
1.92
4
÷2
2
÷2
1
B Brand A costs $
÷2
÷2
÷5
÷5
÷2
Ounces
Price ($)
25
4.50
5
÷5
÷5
1
÷2
per ounce. Brand B costs $
per ounce.
C Which brand is the better buy? Why?
Reflect
1.
Analyze Relationships Describe another method to compare the costs.
Lesson 7.2
187
Calculating Unit Rates
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A unit rate is a rate in which the second quantity is one unit. When the first
quantity in a unit rate is an amount of money, the unit rate is sometimes called
a unit price or unit cost.
EXAMPLE 1
6.4.D
A Gerald pays $90 for 6 yoga classes. What is the cost per class?
Yoga Classes
This month’s special:
6 classes
for $90
$90
Use the information in the problem to write a rate: _______
6 classes
To find the unit rate, divide both quantities in the rate
by the same number so that the second quantity is 1:
÷6
$90
$15
________
= ______
6 classes
1 class
÷6
$15
The unit rate _____
1 class
is the same as
15 ÷ 1 = $15 per class.
Gerald’s yoga classes cost $15 per class.
÷2
B The cost of 2 cartons of milk is $5.50.
What is the unit price?
The unit price is $2.75 per carton of milk.
$5.50
$2.75
________
= _______
1 carton
2 cartons
÷2
÷50
The ship travels 0.4 mile per minute.
÷50
Reflect
2.
Analyze Relationships In all of these problems, how is the unit rate
related to the rate given in the original problem?
YOUR TURN
3. There are 156 players on 13 teams. How many players are on each
team?
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Unit 3
players per team
4. A package of 36 photographs costs $18. What is the cost per
photograph? $
per photograph
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The first quantity
in a unit rate can
be less than 1.
C A cruise ship travels 20 miles in 50 minutes.
20 miles = ________
0.4 mile
__________
How far does the ship travel per minute?
50 minutes 1 minute
Problem Solving with Unit Rates
You can solve rate problems by using a unit rate or by using equivalent rates.
EXAMPL 2
EXAMPLE
6.4.D
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At a summer camp, the campers are divided into groups. Each group has
16 campers and 2 cabins. How many cabins are needed for 112 campers?
Method 1 Find the unit rate. How many campers per cabin?
÷2
16 campers _________
8 campers
__________
=
2 cabins
1 cabin
Divide to find the unit rate.
÷2
There are 8 campers per cabin.
112 campers
_________________
= 14 cabins
8 campers per cabin
Divide to find the number of
cabins.
Method 2 Use equivalent rates.
×7
16 campers ___________
112 campers
__________
=
2 cabins
14 cabins
×7
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Math
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The camp needs 14 cabins.
Reflect
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5.
What If? Suppose each group has 12 campers and 3 canoes. Find the
unit rate of campers to canoes.
YOUR TURN
6.
Petra jogs 3 miles in 27 minutes. At this rate, how long would it take her
to jog 5 miles?
7.
When Jerry drives 100 miles on the highway, his car uses 4 gallons of
gasoline. How much gasoline would his car use if he drives 275 miles
on the highway?
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Lesson 7.2
189
Guided Practice
The sizes and prices of three brands of laundry detergent are shown
in the table. Use the table for 1 and 2. (Explore Activity)
1. What is the unit price for each detergent?
Brand
Size (oz)
Price ($)
A
32
4.80
per ounce
B
48
5.76
per ounce
C
128
17.92
Brand A: $
per ounce
Brand B: $
Brand C: $
2. Which detergent is the best buy?
Mason’s favorite brand of peanut butter is available in two sizes. Each size and
its price are shown in the table. Use the table for 3 and 4. (Explore Activity)
3. What is the unit rate for each size of peanut butter?
Regular: $
per ounce
Family size: $
per ounce
Size (oz)
Price ($)
Regular
16
3.36
Family Size
40
7.60
4. Which size is the better buy?
Find the unit rate. (Example 1)
5. Lisa walked 48 blocks in 3 hours.
6. Gordon types 1,800 words in 25 minutes.
blocks per hour
words per minute
Solve. (Example 2)
8. The cost of 10 oranges is $1.00. What is the cost of 5 dozen oranges?
9. On Tuesday, Donovan earned $11 for 2 hours of babysitting. On Saturday,
he babysat for the same family and earned $38.50. How many hours did
he babysit on Saturday?
?
?
ESSENTIAL QUESTION CHECK-IN
10. How can you use a rate to compare the costs of two boxes of cereal that
are different sizes?
190
Unit 3
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7. A particular frozen yogurt has 75 calories in 2 ounces. How many calories
are in 8 ounces of the yogurt?
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Class
Date
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11. Abby can buy an 8-pound bag of dog food for $7.40 or a 4-pound bag of
the same dog food for $5.38. Which is the better buy?
12. A bakery offers a sale price of $3.50 for 4 muffins. What is the price
per dozen?
Taryn and Alastair both mow lawns. Each charges a flat fee to mow a lawn.
The table shows the number of lawns mowed in the past week, the time
spent mowing lawns, and the money earned.
Number of Lawns
Mowed
Time Spent Mowing
Lawns (in hours)
Money Earned
Taryn
9
7.5
$112.50
Alastair
7
5
$122.50
13. How much does Taryn charge to mow a lawn?
14. How much does Alastair charge to mow a lawn?
15. Who earns more per hour, Taryn or Alastair?
© Houghton Mifflin Harcourt Publishing Company
16. What If? If Taryn and Alastair want to earn an additional $735 each, how
many additional hours will each spend mowing lawns? Explain.
17. Multistep Tomas makes balloon sculptures at a circus. In 180 minutes, he
uses 252 balloons to make 36 identical balloon sculptures.
a. How many minutes does it take to make 1 balloon sculpture?
b. How many balloons are used in one balloon sculpture?
c. What is Tomas’s unit rate for balloons used per minute?
Lesson 7.2
191
18. Quan and Krystal earned the same number of points playing the
same video game. Quan played for 45 minutes and Krystal played for
30 minutes. Whose rate of points earned per minute was higher? Explain.
Mrs. Jacobsen is a music teacher. She wants to order toy instruments online
to give as prizes to her students. The table below shows the prices for
various order sizes.
Whistles
Kazoos
25 items
50 items
80 items
$21.25
$10.00
$36.00
$18.50
$60.00
$27.20
19. What is the highest unit price per kazoo?
20. Persevere in Problem Solving If Mrs. Jacobsen wants to buy the item
with the lowest unit price, what item should she order and how many
of that item should she order?
FOCUS ON HIGHER ORDER THINKING
Work Area
22. Critique Reasoning A 2-pound box of spaghetti costs $2.50. Philip says
2
that the unit cost is ___
2.50 = $0.80 per pound. Explain his error.
23. Look for a Pattern A grocery store sells three different quantities of
sugar. A 1-pound bag costs $1.10, a 2-pound bag costs $1.98, and a
3-pound bag costs $2.85. Describe how the unit cost changes as the
quantity of sugar increases.
192
Unit 3
© Houghton Mifflin Harcourt Publishing Company
21. Draw Conclusions There are 2.54 centimeters in 1 inch. How many
centimeters are there in 1 foot? in 1 yard? Explain your reasoning.
LESSON
7.3
?
Using Ratios and
Rates to Solve
Problems
ESSENTIAL QUESTION
Proportionality—
6.4.B Apply qualitative
and quantitative reasoning
to solve prediction and
comparison of real-world
problems involving ratios
and rates.
How can you use ratios and rates to make comparisons and
predictions?
6.4.B
EXPLORE ACTIVITY 1
Using Tables to Compare Ratios
Anna’s recipe for lemonade calls for 2 cups of lemonade concentrate
and 3 cups of water.
A In Anna’s recipe, the ratio of lemonade concentrate to water is
Use equivalent ratios to complete the table.
2·2
Lemonade Concentrate (c)
2
Water (c)
3
2·
.
Mathematical Processes
What would happen to the
taste of Anna’s lemonade
if she used more cups of
lemonade concentrate to
make the same amount
of lemonade?
2·
4
3·2
Math Talk
9
15
3·3
3·5
Bailey’s recipe calls for 3 cups of lemonade concentrate and 5 cups
of water.
© Houghton Mifflin Harcourt Publishing Company
B In Bailey’s recipe, the ratio of lemonade concentrate to water is
Use equivalent ratios to complete the table.
Lemonade Concentrate (c)
3
Water (c)
5
3·3
3·4
9
12
.
3·
25
5·3
5·
5·
C Find two columns, one in each table, in which the amount of water is
the same. Circle these two columns.
D Whose recipe makes stronger lemonade? How do you know?
10
E Compare the ratios: __
15
9
__
15
2
_
3
3
_
5
Lesson 7.3
193
EXPLORE ACTIVITY 1 (cont’d)
Reflect
1. Explain the Error Marisol makes the following claim: “Bailey’s lemonade
is stronger because it has more lemonade concentrate. Bailey’s lemonade
has 3 cups of lemonade concentrate, and Anna’s lemonade has only
2 cups of lemonade concentrate.” Explain why Marisol is incorrect.
Comparing Ratios
You can use equivalent ratios to solve real-world problems.
Math On the Spot
EXAMPLE 1
6.4.B
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Mathematical Processes
Ruby changed the recipe
and used more dried fruit,
but still produced the same
amount of nut bars. Who
used more chopped
nuts?
STEP 1
Find the ratio of nuts to fruit in the recipe.
4
_
6
STEP 2
18 is a multiple
of 6 and 9, so find
equivalent ratios
with 18 in the
second term.
Find the ratio of nuts to fruit that Tonya used.
6
_
9
STEP 3
4 cups of nuts to 6 cups of fruit
6 cups of nuts to 9 cups of fruit
×3
Find equivalent ratios that
12
have the same second term. _4 = __
6
18
×2
6
_
9
×3
=
12
__
18
×2
12
12
__
= __
18
18
The ratios are the same. So, Tonya used the same ratio of nuts to
fruit that was given in the recipe.
YOUR TURN
2.
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and Intervention
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194
Unit 3
In the science club, there are 2 sixth graders for every 3 seventh graders.
At this year’s science fair, there were 7 projects by sixth graders for
every 12 projects by seventh graders. Is the ratio of sixth graders to
seventh graders in the science club equivalent to the ratio of science
fair projects by sixth graders to projects by seventh graders? Explain.
© Houghton Mifflin Harcourt Publishing Company
Math Talk
A fruit and nut bar recipe calls for 4 cups of chopped nuts and 6 cups of
dried fruit. When Tonya made a batch of these bars, she used 6 cups of
chopped nuts and 9 cups of dried fruit. Did Tonya use the correct ratio
of nuts to fruit?
EXPLORE ACTIVITY 2
6.4.B
Using Rates to Make Predictions
You can represent rates on a double number line to make predictions.
Janet drives from Clarkson to Humbolt in 2 hours. Suppose Janet drives
for 10 hours. If she maintains the same driving rate, can she drive more
than 600 miles? Justify your answer.
Clarkson
Humbolt
112 miles
The double number line shows the number of miles Janet
drives in various amounts of time.
A Explain how Janet’s rate for two hours is
represented on the double number line.
Miles 0
112
224
336
448
Hours 0
2
4
6
8
10
B Describe the relationship between Janet’s rate for two hours and the
other rates shown on the double number line.
© Houghton Mifflin Harcourt Publishing Company
C Complete the number line.
D At this rate, can Janet drive more than 600 miles in 10 hours? Explain.
E How would Janet’s total distance change if she drove for 10 hours at an
increased rate of speed?
Reflect
3. In fifteen minutes, Lena can finish 2 math
homework problems. How many math
problems can she finish in 75 minutes? Use a
double number line to find the answer.
Minutes
0
15
Problems 0
2
Lesson 7.3
195
Guided Practice
1. Celeste is making fruit baskets for her service club to take to a local
hospital. The directions say to fill the boxes using 5 apples for every
6 oranges. Celeste is filling her baskets with 2 apples for every 3 oranges.
(Explore Activity 1)
a. Complete the tables to find equivalent ratios.
Apples
5
Apples
2
Oranges
6
Oranges
3
b. Compare the ratios. Is Celeste using the correct ratio of apples to
oranges?
2. Neha used 4 bananas and 5 oranges in her fruit salad. Daniel used
7 bananas and 9 oranges. Did Neha and Daniel use the same ratio of
bananas to oranges? If not, who used the greater ratio of bananas to
oranges? (Example 1)
Words
0
28
Minutes 0
1
4. A cafeteria sells 30 drinks every 15 minutes. Predict how many drinks the
cafeteria sells every hour. (Explore Activity 2)
?
?
ESSENTIAL QUESTION CHECK-IN
5. Explain how to compare two ratios.
196
Unit 3
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3. Tim is a first grader and reads 28 words per minute. Assuming he
maintains the same rate, use the double number line to find how
many words he can read in 5 minutes. (Explore Activity 2)
Name
Class
Date
7.3 Independent Practice
Personal
Math Trainer
6.4.B
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Online
Assessment and
Intervention
6. Last week, Gina’s art teacher mixed 9 pints of red paint with 6 pints of white
paint to make pink. Gina mixed 4 pints of red paint with 3 pints of white
paint to make pink.
a. Did Gina use the same ratio of red paint to white paint as her teacher? Explain.
b. Yesterday, Gina again mixed red and white paint and made the same
amount of paint, but she used one more pint of red paint than she
used last week. Predict how the new paint color will compare to the
paint she mixed last week.
7. The Suarez family paid $15.75 for 3 movie tickets. How much would they
have paid for 12 tickets?
8. A grocery store sells snacks by weight. A six-ounce bag of mixed nuts
costs $3.60. Predict the cost of a two-ounce bag.
9. The Martin family’s truck gets an average of 25 miles per gallon. Predict
how many miles they can drive using 7 gallons of gas.
10. Multistep The table shows two cell phone plans that offer free minutes for
each given number of paid minutes used. Pablo has Plan A and Sam has Plan B.
© Houghton Mifflin Harcourt Publishing Company
a. What is Pablo’s ratio of free to paid minutes?
b. What is Sam’s ratio of free to paid minutes?
c. Does Pablo’s cell phone plan offer the same ratio of
free to paid minutes as Sam’s? Explain.
Cell Phone Plans
Plan A
Plan B
Free minutes
2
8
Paid minutes
10
25
11. Consumer Math A store has apples on sale for $3.00 for 2 pounds.
a. If an apple is approximately 5 ounces, how many apples can you buy
for $9? Explain.
b. If Dabney paid less per pound for the same number of apples at a
different store, what can you predict about the total cost of the apples?
Lesson 7.3
197
12. Sophie and Eleanor are making bouquets using daisies and tulips. Each
bouquet will have the same total number of flowers. Eleanor uses fewer
daisies in her bouquet than Sophie. Whose bouquet will have the greater
ratio of daisies to total flowers? Explain.
13. A town in east Texas received 10 inches of rain in two weeks. If it kept raining
at this rate for a 31-day month, how much rain did the town receive?
14. One patterned blue fabric sells for $15.00 every two yards, and another sells
for $37.50 every 5 yards. Do these fabrics have the same unit cost? Explain.
Work Area
FOCUS ON HIGHER ORDER THINKING
15. Problem Solving Complete each ratio table.
12
4.5
18
24
18
80.8
40.4
512
10.1
256
17. Analyze Relationships Explain how you can be sure that all the rates you
have written on a double number line are correct.
198
Unit 3
© Houghton Mifflin Harcourt Publishing Company
16. Represent Real-World Problems Write a real-world problem that
compares the ratios 5 to 9 and 12 to 15.
MODULE QUIZ
Ready
Personal
Math Trainer
7.1 Ratios
Online Assessment
and Intervention
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Use the table to find each ratio.
1. white socks to brown socks
2. blue socks to nonblue socks
Color
of socks
Number
of socks
white black blue brown
8
6
4
5
3. black socks to all of the socks
4. Find two ratios equivalent to the ratio in Exercise 1.
7.2 Rates
Find each rate.
5. Earl runs 75 meters in 30 seconds. How many meters does Earl run
per second?
6. The cost of 3 scarves is $26.25. What is the unit price?
7.3 Using Ratios and Rates to Solve Problems
7. Danny charges $35 for 3 hours of swimming lessons. Martin charges
$24 for 2 hours of swimming lessons. Who offers a better deal?
© Houghton Mifflin Harcourt Publishing Company
8. There are 32 female performers in a dance recital. The ratio of men
to women is 3:8. How many men are in the dance recital?
ESSENTIAL QUESTION
9. How can you use ratios and rates to solve problems?
Module 7
199
Personal
Math Trainer
MODULE 7 MIXED REVIEW
Texas Test Prep
1. Which ratio is not equivalent to the other
three?
2
_
A 3
6
B _
9
12
__
15
18
D __
27
C
2. A lifeguard received 15 hours of first aid
training and 10 hours of cardiopulmonary
resuscitation (CPR) training. What is the
ratio of hours of CPR training to hours of
first aid training?
A 15:10
C
B 15:25
D 25:15
10:15
3. Jerry bought 4 DVDs for $25.20. What was
the unit rate?
A $3.15
C
B $4.20
D $8.40
$6.30
4. There are 1,920 fence posts used in a
12-kilometer stretch of fence. How many
fence posts are used in 1 kilometer of fence?
A 150
C
B 160
D 180
155
5. Sheila can ride her bicycle 6,000 meters
in 15 minutes. How far can she ride her
bicycle in 2 minutes?
A 400 meters
C
B 600 meters
D 1,000 meters
800 meters
6. Lennon has a checking account. He
withdrew $130 from an ATM Tuesday.
Wednesday he deposited $240. Friday he
wrote a check for $56. What was the total
change in Lennon’s account?
200
A –$74
C
B $54
D $184
Unit 3
$166
7. Cheyenne is making a recipe that uses
5 cups of beans and 2 cups of carrots.
Which combination below uses the same
ratio of beans to carrots?
A 10 cups of beans and 3 cups of carrots
B 10 cups of beans and 4 cups of carrots
C
12 cups of beans and 4 cups of carrots
D 12 cups of beans and 5 cups of carrots
8. _58 of the 64 musicians in a music contest are
guitarists. Some of the guitarists play jazz
solos, and the rest play classical solos. The
ratio of the number of guitarists playing
jazz solos to the total number of guitarists
in the contest is 1:4. How many guitarists
play classical solos in the contest?
A 10
C
B 20
D 40
30
Gridded Response
9. Mikaela is competing in a race in which
she both runs and rides a bicycle. She
runs 5 kilometers in 0.5 hour and rides her
bicycle 20 kilometers in 0.8 hour. At this
rate, how many kilometers can Mikaela
ride her bicycle in one hour?
.
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
© Houghton Mifflin Harcourt Publishing Company
Selected Response
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Online
Assessment and
Intervention
Applying Ratios
and Rates
?
MODULE
8
LESSON 8.1
ESSENTIAL QUESTION
Comparing Additive
and Multiplicative
Relationships
How can you use ratios and
rates to solve real-world
problems?
6.4.A
LESSON 8.2
Ratios, Rates, Tables,
and Graphs
6.5.A
LESSON 8.3
Solving Problems
with Proportions
6.5.A
LESSON 8.4
Converting
Measurements
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Contributor/Getty Images
6.4.H
Real-World Video
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Chefs use lots of measurements when preparing meals.
If a chef needs more or less of a dish, he can use ratios to
scale the recipe up or down. Using proportional reasoning,
the chef keeps the ratios of all ingredients constant.
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Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your
write-in student
edition, accessible on
any device.
Scan with your smart
phone to jump directly
to the online edition,
video tutor, and more.
Interactively explore
key concepts to see
how math works.
Get immediate
feedback and help as
you work through
practice sets.
201
Are YOU Ready?
Personal
Math Trainer
Complete these exercises to review skills you will need
for this chapter.
Graph Ordered Pairs (First Quadrant)
EXAMPLE
y
To graph A(2, 7), start at the origin.
Move 2 units right.
Then move 7 units up.
Graph point A(2, 7).
10
8
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Online
Assessment and
Intervention
A
6
4
2
O
x
2
4
6
8 10
Graph each point on the coordinate grid above.
1. B(9, 6)
2. C(0, 2)
3. D(6, 10)
4. E(3, 4)
Write Equivalent Fractions
EXAMPLE
28
14
×2
__
_____
= 14
= __
21
21 × 2
42
÷7
14
__
_____
= 14
= _23
21
21 ÷ 7
Multiply the numerator and denominator by the
same number to find an equivalent fraction.
Divide the numerator and denominator by the
same number to find an equivalent fraction.
Write the equivalent fraction.
4 = _____
6. __
6
12
1 = _____
7. __
56
8
5 = _____
25
9. __
9
5 = _____
20
10. __
6
36 = _____
12
11. ___
45
9 = _____
8. ___
12
4
20 = _____
10
12. ___
36
Multiples
EXAMPLE
List the first five multiples of 4.
4×1=4
4×2=8
4 × 3 = 12
4 × 4 = 16
4 × 5 = 20
Multiply 4 by the numbers 1, 2,
3, 4, and 5.
List the first five multiples of each number.
13. 3
202
Unit 3
14. 7
15. 8
© Houghton Mifflin Harcourt Publishing Company
6 = _____
5. __
8
32
Reading Start-Up
Visualize Vocabulary
Use the ✔ words to complete the graphic.
Comparing Unit Rates
Single item
Rate in which the
second quantity is
one unit
Ratio of two quantities that
have different units
Review Words
equivalent ratios (razones
equivalentes)
factor (factor)
graph (gráfica)
✔ pattern (patrón)
point (punto)
✔ rate (tasa)
ratio (razón)
✔ unit (unidad)
✔ unit rate (tasa unitaria)
Preview Words
conversion factor (factor
de conversión)
hypotenuse (hipotenusa)
legs (catetos)
proportion (proporción)
scale drawing (dibujo a
escala)
scale factor (factor de
escala)
Numbers that follow a rule
Understand Vocabulary
Complete the sentences using the preview words.
1. A
equivalent measurements.
Vocabulary
is a rate that compares two
© Houghton Mifflin Harcourt Publishing Company
2. The two sides that form the right angle of a right triangle are
called
. The side opposite the right angle in a
right triangle is called the
.
Active Reading
Tri-Fold Before beginning the module, create
a tri-fold to help you learn the concepts and
vocabulary in this module. Fold the paper into
three sections. Label one column “Rates and
Ratios,” the second column “Proportions,” and
the third column “Converting Measurements.”
Complete the tri-fold with important vocabulary,
examples, and notes as you read the module.
Module 8
203
MODULE 8
Unpacking the TEKS
Understanding the TEKS and the vocabulary terms in the TEKS
will help you know exactly what you are expected to learn in this
module.
6.4.H
Convert units within a
measurement system, including
the use of proportions and unit
rates.
Key Vocabulary
unit rate (tasa unitaria)
A rate in which the second
quantity in the comparison is
one unit.
What It Means to You
You will convert measurements using unit rates.
UNPACKING EXAMPLE 6.4.H
The Washington Monument is about 185
yards tall. This height is almost equal to the
length of two football fields. About how
many feet is this?
3 ft
185 yd · ____
1 yd
185 yd
3 ft
= _____
· ____
1
1 yd
= 555 ft
6.5.A
Represent mathematical and
real-world problems involving
ratios and rates using scale
factors, tables, graphs and
proportions.
Key Vocabulary
ratio (razón)
A comparison of two quantities
by division.
rate (tasa)
A ratio that compares two
quantities measured in
different units.
Visit my.hrw.com
to see all
the
unpacked.
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204
Unit 3
What It Means to You
You will use ratios and rates to solve real-world
problems such as those involving proportions.
UNPACKING EXAMPLE 6.5.A
The distance from Austin to Dallas is about 200 miles. How far
1 in.
apart will these cities appear on a map with the scale of ____
?
50 mi
?
1
___
= __
200
50
? = 4 inches
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Getty Images
The Washington Monument is about 555 feet tall.
LESSON
8.1
?
Comparing Additive
and Multiplicative
Relationships
ESSENTIAL QUESTION
Proportionality—
6.4.A Compare two
rules verbally, numerically,
graphically, and symbolically
in the form of y = ax or
y = x + a in order to
differentiate between
additive and multiplicative
relationships.
How do you represent, describe, and compare additive
and multiplicative relationships?
6.4.A
EXPLORE ACTIVITY
Discovering Additive and
Multiplicative Relationships
A Every state has two U.S. senators. The number of electoral votes
a state has is equal to the total number of U.S. senators and U.S.
representatives.
The number of electoral votes is
the number of representatives.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Medioimages
Photodisc/Getty Images
Complete the table.
Representatives
1
2
Electoral votes
3
4
5
25
41
Describe the rule: The number of electoral votes is equal to
the number of representatives plus / times
.
B Frannie orders three DVDs per month from her DVD club.
Complete the table.
Months
1
2
DVDs ordered
3
6
4
13
22
Describe the rule: The number of DVDs ordered is equal to
the number of months plus / times
.
Reflect
1. Look for a Pattern What operation did you use to complete
the tables in A and B ?
Lesson 8.1
205
Graphing Additive and Multiplicative
Relationships
Math On the Spot
To find the number of electoral votes in part A of the Explore, add 2 to the
number of representatives. We call this an additive relationship.
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To find the number of DVDs Frannie has ordered after a given number of
months, multiply the number of months by 3. We call this a multiplicative
relationship.
EXAMPLE 1
A Jolene is packing her lunch for school. The empty lunch box weighs
five ounces. Graph the relationship between the weight of the items in
Jolene’s lunch and the total weight of the packed lunchbox.
STEP 1
The total weight is equal
to the weight of the
items plus the weight
of the lunchbox. The
relationship is additive.
STEP 2
Make a table relating the weight of the items to the
total weight.
Weight of items (oz)
1
2
3
4
5
Total weight (oz)
6
7
8
9
10
To find the total weight, add the weight of the items
and the weight of the lunchbox.
Total
weight
=
Weight
of items
+
Weight of
lunchbox
9
=
4
+
5
List the ordered pairs from the table.
The ordered pairs are (1, 6), (2, 7), (3, 8), (4, 9), and (5, 10).
STEP 3
Graph the ordered pairs on a coordinate plane.
Total Weight (oz)
To plot (1,6), go
right 1 unit from
the origin and
then up 6 units.
The points of the graph
form a straight line for
an additive relationship.
10
8
6
A line drawn through
the points would not go
through the origin.
4
2
O
2
4
6
8 10
Weight of Items (oz)
206
Unit 3
© Houghton Mifflin Harcourt Publishing Company
My Notes
6.4.A
B Oskar sells bracelets for two dollars
each and donates the money he collects
to a charity. Graph the relationship
between the number of bracelets sold
and the total donation.
STEP 1
Complete the table.
Bracelets sold
1
2
3
4
5
Total donation ($)
2
4
6
8
10
To find the total donation, multiply the number of
bracelets sold by the donation per bracelet.
STEP 2
Total
donation
=
Bracelets
sold
×
Donation
per bracelet
10
=
5
×
2
His donation is equal to the
number of bracelets sold
times the donation for each
bracelet. The relationship is
multiplicative.
List the ordered pairs from the table.
The ordered pairs are (1, 2), (2, 4), (3, 6), (4, 8), and (5, 10).
STEP 3
Graph the ordered pairs on a coordinate plane.
The points of the graph form a
straight line for a multiplicative
pattern.
Donation ($)
10
8
6
A line drawn through the points
would intersect the origin.
4
2
O
2
4
6
The line is steeper than the line
in part A.
8 10
Math Talk
Mathematical Processes
How are the graphs
in part A and part B the
same? How are they
different?
YOUR TURN
2. Ky is seven years older than his sister Lu.
Graph the relationship between Ky’s age
and Lu’s age. Is the relationship additive or
multiplicative? Explain.
Lu’s age
Ky’s age
1
2
3
4
5
12
Ky’s Age (years)
© Houghton Mifflin Harcourt Publishing Company
Bracelets Sold
10
8
6
4
2
O
2
4
6
8 10
Lu’s Age (years)
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Math Trainer
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Lesson 8.1
207
Guided Practice
Dogs adopted
1
2
3
2. Graph the relationship between the
number of dogs adopted and the total
number of dogs. (Example 1)
Number of Dogs
1. Fred’s family already has two dogs. They
adopt more dogs. Complete the table for
the total number of dogs they will have.
Then describe the rule. (Explore Activity)
4
Total number
of dogs
10
8
6
4
2
O
2
6
4
8 10
Dogs Adopted
3. Frank’s karate class meets three days every
week. Complete the table for the total
number of days the class meets. Then
describe the rule. (Explore Activity)
4. Graph the relationship between the
number of weeks and the number of days
of class. (Example 1)
1
2
3
4
Days of
class
Days of class
30
Weeks
24
18
12
6
O
2
4
6
8 10
0.50
0.40
0.30
0.20
0.10
O
2
4
6
8 10
Pages Printed
?
?
ESSENTIAL QUESTION CHECK-IN
6. How do you represent, describe, and compare additive
and multiplicative relationships?
208
Unit 3
© Houghton Mifflin Harcourt Publishing Company
5. An internet café charges ten cents for each page printed.
Graph the relationship between the number of pages
printed and the printing charge. Is the relationship
additive or multiplicative? Explain. (Example 1)
Printing Charges ($)
Weeks
Name
Class
Date
8.1 Independent Practice
Personal
Math Trainer
6.4.A
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The tables give the price of a kayak rental
from two different companies.
Online
Assessment and
Intervention
The graph represents the distance traveled by
a car and the number of hours it takes.
Raging River Kayaks
1
3
6
8
Cost ($)
9
27
54
72
Paddlers
Hours
2
4
5
10
Cost ($)
42
44
45
50
7. Is the relationship shown in each table
multiplicative or additive? Explain.
© Houghton Mifflin Harcourt Publishing Company
8. Yvonne wants to rent a kayak for 7 hours.
How much would this cost at each
company? Which one should she choose?
9. After how many hours is the cost for both
kayak rental companies the same? Explain
how you found your answer.
Distance (mi)
600
Hours
480
360
240
120
O
2
4
6
8
10
Time (h)
10. Persevere in Problem Solving Based
on the graph, was the car traveling at a
constant speed? At what speed was the
car traveling?
11. Make a Prediction If the pattern shown
in the graph continues, how far will the car
have traveled after 6 hours? Explain how
you found your answer.
12. What If? If the car had been traveling at
40 miles per hour, how would the graph be
different?
Lesson 8.1
209
Use the graph for Exercises 13–15.
13. Which set of points represents an additive relationship? Which set of
points represents a multiplicative relationship?
24
20
16
12
14. Represent Real-World Problems What is a real-life relationship that
might be described by the red points?
8
4
O
1
2
3
4
5
15. Represent Real-World Problems What is a real-life relationship that
might be described by the black points?
Work Area
FOCUS ON HIGHER ORDER THINKING
16. Explain the Error An elevator
Time (s)
leaves the ground floor and rises
Distance (ft)
three feet per second. Lili makes
the table shown to analyze the
relationship. What error did she make?
1
2
3
4
4
5
6
7
17. Analyze Relationships Complete each table. Show an additive
relationship in the first table and a multiplicative relationship in the
second table.
B
1
2
3
A
1
2
B
16
32
3
Use two columns of each table. Which table shows equivalent ratios?
Name two ratios shown in the table that are equivalent.
18. Represent Real-World Problems Describe a real-world situation
that represents an additive relationship and one that represents a
multiplicative relationship.
210
Unit 3
© Houghton Mifflin Harcourt Publishing Company
A
LESSON
8.2
?
Ratios, Rates, Tables,
and Graphs
ESSENTIAL QUESTION
Proportionality—
6.5.A Represent
mathematical and real-world
problems involving ratios
and rates using … tables,
graphs, …
How can you represent real-world problems involving ratios and
rates with tables and graphs?
6.5.A
EXPLORE ACTIVITY 1
Finding Ratios from Tables
Students in Mr. Webster’s science classes are doing an experiment that requires
250 milliliters of distilled water for every 5 milliliters of solvent. The table shows
the amount of distilled water needed for various amounts of solvent.
Solvent (mL)
2
Distilled water (mL)
3
3.5
100
5
200
250
A Use the numbers in the first column of the table to write a ratio of
distilled water to solvent.
B How much distilled water is used for 1 milliliter of solvent?
Use your answer to write another ratio of distilled water to solvent.
© Houghton Mifflin Harcourt Publishing Company
C The ratios in
A
and
B
are equivalent/not equivalent.
D How can you use your answer to B to find the amount of distilled
water to add to a given amount of solvent?
Math Talk
Mathematical Processes
Is the relationship between
the amount of solvent and
the amount of distilled water
additive or multiplicative?
Explain.
E Complete the table. What are the equivalent ratios shown in the table?
100 = _____ = _____ = _____
250
200 = ____
____
2
3
3.5
5
Reflect
1.
Look for a Pattern When the amount of solvent increases
by 1 milliliter, the amount of distilled water increases by
milliliters. So 6 milliliters of solvent requires
distilled water.
milliliters of
Lesson 8.2
211
6.5.A
EXPLORE ACTIVITY 2
Graphing with Ratios
A Copy the table from Explore Activity 1 that shows the amounts of
solvent and distilled water.
Distilled
water (mL)
2
3
100
3.5
5
200
250
B Write the information in the table as ordered pairs. Use the
amount of solvent as the x-coordinates and the amount of
distilled water as the y-coordinates.
(2,
) (3,
), (3.5,
), (
, 200), (5, 250)
Graph the ordered pairs and connect the points.
Distilled Water (mL)
Solvent (mL)
(5, 250)
300
200
100
O
2
4
6
Solvent (mL)
Describe your graph.
C For each ordered pair that you graphed, write the ratio of the
y-coordinate to the x-coordinate.
D The ratio of distilled water to solvent is _____
. How are the ratios in
1
C
related to this ratio?
E The point (2.5, 125) is on the graph but not in the table. The ratio of the
y-coordinate to the x-coordinate is
C
and
D
?
2.5 milliliters of solvent requires
milliliters of distilled water.
F Conjecture What do you think is true for every point on the graph?
Reflect
2.
212
Communicate Mathematical Ideas How can you use the graph to find
the amount of distilled water to use for 4.5 milliliters of solvent?
Unit 3
© Houghton Mifflin Harcourt Publishing Company
the ratios in
. How is this ratio related to
Representing Rates with
Tables and Graphs
You can use tables and graphs to represent real-world problems involving
equivalent rates.
Math On the Spot
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EXAMPL 1
EXAMPLE
6.5.A
The Webster family is taking an
express train to Washington, D.C. The
train travels at a constant speed and
makes the trip in 2 hours.
Animated
Math
A Make a table to show the distance
the train travels in various amounts
of time.
STEP 1
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Write a ratio of distance
to time to find the rate.
distance = _________
120 miles = _______
60 miles = 60 miles per hour
________
time
STEP 2
2 hours
1 hour
Use the unit rate to make a table.
Time (h)
Distance (mi)
2
3
3.5
4
5
120
180
210
240
300
B Graph the information from the table.
(2, 120), (3, 180), (3.5, 210), (4, 240),
(5, 300)
Distance (mi)
Write ordered pairs. Use Time as the
x-coordinates and Distance as the
y-coordinates.
O
(2, 120)
x
1 2 3 4 5
Time (h)
STEP 2
Graph the ordered pairs and connect the points.
YOUR TURN
3. A shower uses 12 gallons of water in 3 minutes.
Complete the table and graph.
Time (min)
Water used (gal)
2
3
3.5
6.5
20
Water used (gal)
© Houghton Mifflin Harcourt Publishing Company
STEP 1
y
300
240
180
120
60
40
32
24
16
8
O
Personal
Math Trainer
2
4
6
8 10
Time (min)
Online Assessment
and Intervention
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Lesson 8.2
213
Guided Practice
Sulfur atoms
6
9
2. Graph the relationship between sulfur atoms
and oxygen atoms. (Explore Activity 2)
Oxygen Atoms
1. Sulfur trioxide molecules all have the same
ratio of oxygen atoms to sulfur atoms.
A number of molecules of sulfur dioxide
have 18 oxygen atoms and 6 sulfur atoms.
Complete the table. (Explore Activity 1)
21
Oxygen
atoms
81
90
72
54
36
18
O
What are the equivalent ratios shown in the
table?
3. Stickers are made with the same ratio of
width to length. A sticker 2 inches wide has
a length of 4 inches. Complete the table.
(Explore Activity 1)
6 12 18 24 30
Sulfur Atoms
4. Graph the relationship between the width
and the length of the stickers. (Explore
Activity 2)
2
4
7
Length (in.)
16
What are the equivalent ratios shown in the
table?
Length (in.)
20
Width (in.)
16
12
8
4
O
2
4
6
8 10
Width (in.)
Candles
5
8
120
96
72
48
24
O
?
?
2
4
ESSENTIAL QUESTION CHECK-IN
6. How do you represent real-world problems involving ratios and rates with
tables and graphs?
214
6
Boxes
Unit 3
8 10
© Houghton Mifflin Harcourt Publishing Company
Boxes
120
Candles
5. Five boxes of candles contain a total of 60 candles.
Each box holds the same number of candles. Complete
the table and graph the relationship. (Example 1)
Name
Class
Date
8.2 Independent Practice
Personal
Math Trainer
6.5.A
my.hrw.com
Online
Assessment and
Intervention
The table shows information about the number of sweatshirts sold and the
money collected at a fundraiser for school athletic programs. For Exercises
7–12, use the table.
Sweatshirts sold
3
Money collected ($)
60
5
8
12
180
7. Find the rate of money collected per sweatshirt sold. Show your work.
8. Use the unit rate to complete the table.
9. Explain how to graph information from the table.
11. What If? How much money would be collected if
24 sweatshirts were sold? Show your work.
Money Collected ($)
© Houghton Mifflin Harcourt Publishing Company
10. Write the information in the table as ordered pairs. Graph the
relationship from the table.
280
240
200
160
120
80
40
O
2
4
6
8 10 12 14
Sweatshirts Sold
12. Analyze Relationships Does the point (5.5, 110) make sense in
this context? Explain.
Lesson 8.2
215
13. Communicate Mathematical Ideas The table
shows the distance Randy drove on one day of
her vacation. Find the distance Randy would have
gone if she had driven for one more hour at the
same rate. Explain how you solved the problem.
Time (h)
1
2
3
4
5
Distance (mi)
55
110
165
220
275
Use the graph for Exercises 14–15.
14. Analyze Relationships Does the relationship show a ratio or a rate? Explain.
15. Represent Real-World Problems What is a real-life relationship that
might be described by the graph?
Time (days)
70
56
42
28
14
O
2
4
6
8 10
Time (weeks)
FOCUS ON HIGHER ORDER THINKING
16. Make a Conjecture Complete the table.
distance
time
Then find the rates ______
and ______
.
time
distance
Time (min)
Distance (m)
1
2
Work Area
distance
_______
=
time
5
25
100
time
_______
=
distance
b. Suppose you graph the points (time, distance) and your friend graphs
(distance, time). How will your graphs be different?
17. Communicate Mathematical Ideas To graph a rate or ratio from a table,
how do you determine the scales to use on each axis?
216
Unit 3
© Houghton Mifflin Harcourt Publishing Company
time
a. Are the ______
rates equivalent? Explain.
distance
Solving Problems
with Proportions
LESSON
8.3
?
Proportionality—
6.5.A Represent
mathematical and real-world
problems involving ratios and
rates using … proportions.
ESSENTIAL QUESTION
How can you solve problems with proportions?
Using Equivalent Ratios
to Solve Proportions
A proportion is a statement that two ratios or rates are equivalent.
1
_
and _26 are equivalent ratios.
3
Math On the Spot
1 _
_
= 2 is a proportion.
3 6
EXAMPL 1
EXAMPLE
my.hrw.com
6.5.A
Sheldon and Leonard are partners in a business. Sheldon makes $2 in
profits for every $5 that Leonard makes. If Leonard makes $20 profit on
the first item they sell, how much profit does Sheldon make?
STEP 1
Sheldon’s profit
______________
Leonard’s profit
STEP 2
$2 ____
___
=
$5
$20
Sheldon’s profit
______________
Leonard’s profit
Use common denominators to write equivalent ratios.
$2
× 4 ____
______
= $5 × 4 $20
© Houghton Mifflin Harcourt Publishing Company
Sheldon’s profit is unknown.
Write a proportion.
$8
____
= ____
$20
$20
20 is a common denominator.
Equivalent ratios with the
same denominators have
the same numerators.
Math Talk
Mathematical Processes
How do you know
8
__
= _2 is a proportion?
20
5
= $8
If Leonard makes $20 profit, Sheldon makes $8 profit.
YOUR TURN
1.
The PTA is ordering pizza for their next meeting. They plan to order
2 cheese pizzas for every 3 pepperoni pizzas they order. How many
cheese pizzas will they order if they order 15 pepperoni pizzas?
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Math Trainer
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Lesson 8.3
217
Using Unit Rates to Solve Proportions
You can also use equivalent rates to solve proportions. Finding a unit rate may
help you write equivalent rates.
Math On the Spot
EXAMPLE 2
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6.5.A
The distance Ali runs in 36 minutes is shown on the pedometer.
At this rate, how far could he run in 60 minutes?
My Notes
STEP 1
Write a proportion.
time
________
36 minutes
__________
distance
3 miles
60 minutes
= __________
miles
time
________
distance
60 is not a multiple of 36.
STEP 2
Find the unit rate of the rate you know.
36
÷ 3 = ___
12
______
1
3÷3
60 minutes
12 minutes = __________
__________
You know that Ali runs
3 miles in 36 minutes.
1 mile
STEP 3
miles
Write equivalent rates.
Think: You can multiply 12 × 5 = 60. So multiply the
Math Talk
Compare the fractions
36
60
__
and __
5 using <, >
3
or =. Explain.
12
× 5 = ___
60
______
1×5
60 = ___
60
___
5
Equivalent rates
with the same
numerators
have the same
denominators.
= 5 miles
At this rate, Ali can run 5 miles in 60 minutes.
YOUR TURN
2.
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Math Trainer
Ms. Reynold’s sprinkler system has 9 stations that water all the parts
of her front and back lawn. Each station runs for an equal amount of
time. If it takes 48 minutes for the first 4 stations to water, how long
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and Intervention
does it take to water all parts of her lawn?
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218
Unit 3
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denominator by the same number.
Mathematical Processes
Using Proportional Relationships
to Find Distance on a Map
A scale drawing is a drawing of a real object that is proportionally smaller or
larger than the real object. A scale factor is a ratio that describes how much
smaller or larger the scale drawing is than the real object.
Math On the Spot
my.hrw.com
A map is a scale drawing. The measurements on a map are in proportion to
the actual distance. If 1 inch on a map equals 2 miles actual distance, the scale
2 miles
factor is ______
.
1 inch
EXAMPL 3
EXAMPLE
6.5.A
miles
2 miles = ________
______
1 inch
3 inches
rk
Pa
The scale factor is a unit rate.
d.
Blv
R
Use common denominators to write equivalent ratios.
Scale: 1 inch = 2 miles
2
× 3 = ___
_____
1×3
3
3 is a common denominator.
6 miles = _______
_______
3 inches
T
3 in.
Lehigh Ave.
STEP 2
Eighth St.
Write a proportion.
Broad St.
STEP 1
North St.
The distance between two schools on Lehigh Avenue is shown on the
map. What is the actual distance between the schools?
3 inches
Equivalent ratios with the same
denominators have the same numerators.
= 6 miles
© Houghton Mifflin Harcourt Publishing Company
The actual distance between the two schools is 6 miles.
YOUR TURN
3.
The distance between
Sandville and Lewiston
is shown on the map.
What is the actual
distance between the
towns?
Sandville
Traymoor
2.5 in.
Sloneham
Lewiston
Baymont
Scale: 1 inch = 20 miles
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Lesson 8.3
219
Guided Practice
Find the unknown value in each proportion. (Example 1)
3 = ___
1. __
5 30
4 = ___
2. ___
5
10
4÷
__________
= _____
3
×
_________
= _____
30
5×
10 ÷
5
Solve using equivalent ratios. (Example 1)
3. Leila and Jo are two of the partners in
a business. Leila makes $3 in profits
for every $4 that Jo makes. If Jo makes
$60 profit on the first item they sell, how
4. Hendrick wants to enlarge a photo that
is 4 inches wide and 6 inches tall. The
enlarged photo keeps the same ratio.
How tall is the enlarged photo if it is
much profit does Leila make?
12 inches wide?
5. A person on a moving sidewalk travels
21 feet in 7 seconds. The moving
sidewalk has a length of 180 feet. How
long will it take to move from one end
to the other?
6. In a repeating musical pattern, there
are 56 beats in 7 measures. How many
measures are there after 104 beats?
7. Contestants in a dance-a-thon rest for
the same amount of time every hour.
A couple rests for 25 minutes in 5 hours.
How long did they rest in 8 hours?
8. Francis gets 6 paychecks in 12 weeks.
How many paychecks does she get in
52 weeks?
9. What is the actual distance between Gendet
and Montrose?
?
?
(Example 3)
ESSENTIAL QUESTION CHECK-IN
10. How do you solve problems with proportions?
Gravel
Gendet
1.5 cm
Montrose
Scale: 1 centimeter = 16 kilometers
220
Unit 3
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Solve using unit rates. (Example 2)
Name
Class
Date
8.3 Independent Practice
Personal
Math Trainer
6.5.A
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11. On an airplane, there are two seats on the
left side in each row and three seats on the
right side. There are 90 seats on the right
side of the plane.
a. How many seats are on the left side of
the plane?
Online
Assessment and
Intervention
a. How many cups of punch does the
recipe make?
b. If Wendell makes 108 cups of punch,
how many cups of each ingredient will
he use?
cups pineapple juice
b. How many seats are there
cups orange juice
altogether?
cups lemon-lime soda
12. The scale of the map is missing. The actual
distance from Liberty to West Quall is 72
miles, and it is 6 inches on the map.
West Quall
Abbeville
Foston
Mayne
Liberty
c. How many servings can be made from
108 cups of punch?
14. Carlos and Krystal are taking a road trip
from Greenville to North Valley. Each has
their own map, and the scales on their
maps are different.
a. On Carlos’s map, Greenville and North
Valley are 4.5 inches apart. The scale on
his map is 1 inch = 20 miles. How far is
Greenville from North Valley?
© Houghton Mifflin Harcourt Publishing Company
a. What is the scale of the map?
b. Foston is directly between Liberty and
West Quall and is 4 inches from Liberty
on the map. How far is Foston from
West Quall? Explain.
b. The scale on Krystal’s map is 1 inch =
18 miles. How far apart are Greenville
and North Valley on Krystal’s map?
15. Multistep A machine can produce 27
inches of ribbon every 3 minutes. How
many feet of ribbon can the machine make
in one hour? Explain.
13. Wendell is making punch for a party.
The recipe he is using says to mix 4 cups
pineapple juice, 8 cups orange juice, and
12 cups lemon-lime soda in order to make
18 servings of punch.
Lesson 8.3
221
Marta, Loribeth, and Ira all have bicycles.
The table shows the number of miles of
each rider’s last bike ride, as well as the
time it took each rider to complete the ride.
16. What is Marta’s unit rate, in minutes per
Distance of Last Time Spent on Last Bike
Ride (in miles)
Ride (in minutes)
Marta
8
80
Loribeth
6
42
Ira
15
75
mile?
17. Whose speed was the fastest on their last bike ride?
18. If all three riders travel for 3.5 hours at the same speed as their last ride,
how many total miles will all 3 riders have traveled? Explain.
19. Critique Reasoning Jason watched a caterpillar move 10 feet in
2 minutes. Jason says that the caterpillar’s unit rate is 0.2 feet per minute.
Is Jason correct? Explain.
Work Area
FOCUS ON HIGHER ORDER THINKING
21. Multiple Representations A boat travels at a constant speed. After
20 minutes, the boat has traveled 2.5 miles. The boat travels a total of
10 miles to a bridge.
b. How long does it take the boat to reach
the bridge? Explain how you found it.
Distance (mi)
10
a. Graph the relationship between the
distance the boat travels and the time
it takes.
8
6
4
2
O
20
60
Time (min)
222
Unit 3
100
© Houghton Mifflin Harcourt Publishing Company
20. Analyze Relationships If the number in the numerator of a unit rate is 1,
what does this indicate about the equivalent unit rates? Give an example.
LESSON
8.4
?
Converting
Measurements
Proportionality—
6.4.H Convert units
within a measurement
system, including the use of
proportions and unit rates.
ESSENTIAL QUESTION
How do you convert units within a measurement system?
6.4.H
EXPLORE ACTIVITY
Using a Model to Convert Units
The two most common systems of measurement are the customary system and
the metric system. You can use a model to convert from one unit to another
within the same measurement system.
STEP 1
Use the model to complete each
statement below.
1 yard = 3 feet
STEP 2
6
9
12
1
2
3
4
feet
yards
2 yards =
feet
3 yards =
feet
4 yards =
feet
Write each rate you found in Step 1 in simplest form.
6 feet
3 feet
______
= _______
2 yards
1 yard(s)
© Houghton Mifflin Harcourt Publishing Company
3
9 feet
3 feet
______
= _______
3 yards
1 yard(s)
12 feet
3 feet
______
= _______
4 yards
1 yard(s)
Since 1 yard = 3 feet, the rate of feet to yards in any measurement
is always _31. This means any rate forming a proportion with _31 can
represent a rate of feet to yards.
3 __
_
= 12
, so 12 feet =
1
4
yards.
3 __
_
= 54, so
1 18
feet = 18 yards.
Reflect
1.
Communicate Mathematical Ideas How could you draw a model to
show the relationship between feet and inches?
Lesson 8.4
223
Converting Units Using
Proportions and Unit Rates
Math On the Spot
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You can use rates and proportions to convert both customary and metric units.
Use the table below to convert from one unit to another within the same
measurement system.
Customary Measurements
Length
Weight
1 ft = 12 in.
1 yd = 36 in.
1 yd = 3 ft
1 mi = 5,280 ft
1 mi = 1,760 yd
1 lb = 16 oz
1 T = 2,000 lb
Capacity
1 c = 8 fl oz
1 pt = 2 c
1 qt = 2 pt
1 qt = 4 c
1 gal = 4 qt
Metric Measurements
Length
Mass
1 km = 1,000 m
1 m = 100 cm
1 cm = 10 mm
1 kg = 1,000 g
1 g = 1,000 mg
Capacity
1 L = 1,000 mL
EXAMPLE 1
My Notes
6.4.H
A What is the weight of a 3-pound human brain in ounces?
Use a proportion to convert 3 pounds to ounces.
16 ounces
Use _______
to convert pounds to ounces.
1 pound
STEP 1
Write a proportion.
16 ounces = _________
ounces
_________
STEP 2
3 pounds
Use common denominators to write equivalent ratios.
16
× 3 = ___
______
1×3
3
48 = ___
___
3
3
3 is a common denominator.
Equivalent rates with the same denominators
have the same numerators.
= 48 ounces
The weight is 48 ounces.
B A moderate amount of daily sodium consumption is 2,000 milligrams.
What is this mass in grams?
Use a proportion to convert 2,000 milligrams to grams.
1,000 mg
to convert milligrams to grams.
Use _______
1g
224
Unit 3
© Houghton Mifflin Harcourt Publishing Company
1 pound
STEP 1
Write a proportion.
1,000 mg ________
2,000 mg
________
=
1g
STEP 2
g
Write equivalent rates.
Think: You can multiply 1,000 × 2 = 2,000. So multiply the
denominator by the same number.
2,000
1,000
× 2 _____
_________
=
1×2
2,000
2,000 _____
_____
=
Equivalent rates with the same numerators
have the same denominators.
2
= 2 grams
Math Talk
Mathematical Processes
The mass is 2 grams.
How would you
convert 3 liters to
milliliters?
YOUR TURN
2.
The height of a doorway is 2 yards. What is the height of the doorway
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in inches?
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and Intervention
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Converting Units by Using
Conversion Factors
© Houghton Mifflin Harcourt Publishing Company
Another way to convert measurements is by using a conversion factor.
A conversion factor is a rate comparing two equivalent measurements.
Math On the Spot
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EXAMPL 2
EXAMPLE
6.4.H
Elena wants to buy 2 gallons of milk but can only find quart containers
for sale. How many quarts does she need?
You are converting to
quarts from gallons.
STEP 1 Find the conversion factor.
4 quarts
Write 4 quarts = 1 gallon as a rate: ______
1 gallon
STEP 2
Multiply the given measurement by the conversion factor.
4 quarts
2 gallons · ______
=
1 gallon
quarts
4 quarts
= 8 quarts Cancel the common unit.
2 gallons · ______
1 gallon
Elena needs 8 quarts of milk.
Lesson 8.4
225
YOUR TURN
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3. An oak tree is planted when it is 250 centimeters tall. What is this height
in meters?
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Guided Practice
Use the model below to complete each statement. (Explore Activity 1)
4
8
12
16
1
2
3
4
cups
quarts
12
1. _41 = __
3 , so 12 cups =
quarts
48
2. _41 = __
12, so
cups = 12 quarts
3. Mary Catherine makes 2 gallons of punch
for her party. How many cups of punch did
she make?
4. An African elephant weighs 6 tons. What is
the weight of the elephant in pounds?
5. The distance from Jason’s house to school
is 0.5 kilometer. What is this distance in
meters?
6. The mass of a moon rock is 3.5 kilograms.
What is the mass of the moon rock in grams?
Use a conversion factor to solve. (Example 2)
1,000 mg
7. 1.75 grams · _______
=
1g
9. A package weighs 96 ounces. What is the
weight of the package in pounds?
?
?
1 cm
8. 27 millimeters · ______
=
10 mm
10. A jet flies at an altitude of 52,800 feet.
What is the height of the jet in miles?
ESSENTIAL QUESTION CHECK-IN
11. How do you convert units within a measurement system?
226
Unit 3
© Houghton Mifflin Harcourt Publishing Company
Use unit rates to solve. (Example 1)
Name
Class
Date
8.4 Independent Practice
Personal
Math Trainer
6.4.H
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Online
Assessment and
Intervention
12. What is a conversion factor that you can use to convert gallons to pints?
How did you find it?
13. Three friends each have some ribbon. Carol has 42 inches of ribbon, Tino
has 2.5 feet of ribbon, and Baxter has 1.5 yards of ribbon. Express the total
length of ribbon the three friends have in inches, feet and yards.
inches =
feet =
yards
14. Suzanna wants to measure a board, but she doesn’t have a ruler to
measure with. However, she does have several copies of a book that she
knows is 17 centimeters tall.
a. Suzanna lays the books end to end and finds that the board is the
same length as 21 books. How many centimeters long is the board?
b. Suzanna needs a board that is at least 3.5 meters long. Is the board
long enough? Explain.
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Sheldon needs to buy 8 gallons of ice cream for a family reunion. The table
shows the prices for different sizes of two brands of ice cream.
Price of small size
Price of large size
Cold Farms
$2.50 for 1 pint
$4.50 for 1 quart
Sweet Dreams
$4.25 for 1 quart
$9.50 for 1 gallon
15. Which size container of Cold Farm ice cream is the better deal for
Sheldon? Explain.
16. Multistep Which size and brand of ice cream is the best deal?
Lesson 8.4
227
17. In Beijing in 2008, the Women's 3,000 meter Steeplechase became an
Olympic event. What is this distance in kilometers?
18. How would you convert 5 feet 6 inches to inches?
FOCUS ON HIGHER ORDER THINKING
19. Analyze Relationships A Class 4 truck weighs between 14,000 and
16,000 pounds.
a. What is the weight range in tons?
b. If the weight of a Class 4 truck is increased by 2 tons, will it still be
classified as a Class 4 truck? Explain.
Work Area
20. Persevere in Problem Solving A football field is shown at right.
1
53 3 yd
120 yd
b. A chalk line is placed around the
perimeter of the football field. What is the length of this line in feet?
c. About how many laps around the perimeter of the field would equal
1 mile? Explain.
21. Look for a Pattern What is the result if you multiply a number of cups
1 cup
8 ounces
______
by ______
1 cup and then multiply the result by 8 ounces? Give an example.
22. Make a Conjecture 1 hour = 3,600 seconds and 1 mile = 5,280 feet.
Make a conjecture about how you could convert a speed of 15 miles per
hour to feet per second. Then convert.
228
Unit 3
© Houghton Mifflin Harcourt Publishing Company Image credits: ©Michael Steele/
Getty Images
a. What are the dimensions of
a football field in feet?
MODULE QUIZ
Ready
Personal
Math Trainer
8.1 Comparing Additive and Multiplicative Relationships
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Complete each table and describe the rule for the relationship.
1.
2.
Meal time
12:00
Swim time
12:45
Sets of pens
12:30
Online Assessment
and Intervention
1:00
2:15
2
3
Number of pens
4
5
9
15
8.2 Ratios, Rates, Tables, and Graphs
3. Charlie runs laps around a track. The table shows how long it takes him to
run different numbers of laps. How long would it take Charlie to run 5 laps?
Number of laps
2
4
6
8
10
Time (min)
10
20
30
40
50
8.3 Solving Problems with Proportions
4. Emily is entering a bicycle race for charity. Her mother pledges $0.40
for every 0.25 mile she bikes. If Emily bikes 15 miles, how much will her
© Houghton Mifflin Harcourt Publishing Company
mother donate?
8.4 Converting Measurements
Convert each measurement.
5. 18 meters =
7. 6 quarts =
centimeters
fluid ounces
6. 5 pounds =
8. 9 liters =
ounces
milliliters
ESSENTIAL QUESTION
9. Write a real-world problem that could be solved using a proportion.
Module 8
229
Personal
Math Trainer
MODULE 8 MIXED REVIEW
Texas Test Prep
Selected Response
1. The table below shows the number of
babies and adults at a nursery.
Babies
8
12
16
20
Adults
2
3
4
5
Online
Assessment and
Intervention
my.hrw.com
4. The table below shows the number of
petals and leaves for different numbers of
flowers.
Petals
5
10
15
20
Leaves
2
4
6
8
Which represents the number of babies?
How many petals are present when there
are 12 leaves?
A adults × 6
A 25 petals
B adults × 4
B 30 petals
C
adults + 4
C
D adults + 6
D 36 petals
2. The graph represents the distance Manuel
walks over several hours.
10
Distance (mi)
35 petals
5. A recipe calls for 3 cups of sugar and
9 cups of water. If the recipe is reduced,
how many cups of water should be used
with 2 cups of sugar?
8
6
A 3 cups
4
B 4 cups
2
C
O
2
4
6
6 cups
D 8 cups
8 10
Time (h)
A (2.5, 14)
C
B (1.25, 5)
D (1.5, 9)
(2.25, 12)
3. On a map of the city, 1 inch represents
1.5 miles. What distance on the map would
represent 12 miles?
6. Janice bought 4 oranges for $3.40. What is
the unit price?
.
0
0
0
0
0
0
1
1
1
1
1
1
A 6 inches
2
2
2
2
2
2
B 8 inches
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
C
12 inches
D 18 inches
230
Gridded Response
Unit 3
© Houghton Mifflin Harcourt Publishing Company
Which is an ordered pair on the line?
Percents
?
MODULE
ESSENTIAL QUESTION
How can you use percents
to solve real-world
problems?
9
LESSON 9.1
Understanding
Percent
6.4.E. 6.4.F
LESSON 9.2
Percents, Fractions,
and Decimals
6.4.G, 6.5.C
LESSON 9.3
Solving Percent
Problems
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Digital Vision/
Getty Images
6.5.B, 6.5.G
Real-World Video
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When you eat at a restaurant, your bill will include
sales tax for most items. It is customary to add a tip
for your server in many restaurants. Both taxes and
tips are calculated as a percent of the bill.
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Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your
write-in student
edition, accessible on
any device.
Scan with your smart
phone to jump directly
to the online edition,
video tutor, and more.
Interactively explore
key concepts to see
how math works.
Get immediate
feedback and help as
you work through
practice sets.
231
Are YOU Ready?
Personal
Math Trainer
Complete these exercises to review skills you will need
for this chapter.
Write Equivalent Fractions
EXAMPLE
9
__
=
12
9
__
=
12
36
9×4
_____
= __
12 × 4
48
9÷3
_____
_
= 34
12 ÷ 3
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Online
Assessment and
Intervention
Multiply the numerator and denominator by the same
number to find an equivalent fraction.
Divide the numerator and denominator by the same
number to find an equivalent fraction.
Write the equivalent fraction.
9 = _____
1. ___
6
18
4 = _____
2. __
6
18
25 = _____
5
3. ___
30
36
12 = _____
4. ___
15
15 = _____
5. ___
24
8
24 = _____
6. ___
32
8
50 = _____
10
7. ___
60
5 = _____
20
8. __
9
Multiply Fractions
EXAMPLE
15
13
5
3
__
__
__ × __
×
=
12
10
12
10
4
2
1
_
=8
Divide by the common factors.
Simplify.
9.
3
4
_
× __
=
11
8
9
12. __
× _4 =
20 5
8
10. __
× _5 =
15 6
3
7
11. __
× __
=
12 14
20
7
13. __
× __
=
10 21
8
9
14. __
× __
=
18 20
Decimal Operations (Multiplication)
EXAMPLE
1.6
×0.3
0.48
Multiply as you would with whole numbers.
Count the total number of decimal places in the factors.
Place the decimal point that number of places in the product.
Multiply.
232
Unit 3
15. 20 × 0.25
16. 0.3 × 16.99
17. 0.2 × 75
18. 5.5 × 1.1
19. 11.99 × 0.8
20. 7.25 × 0.5
21. 4 × 0.75
22. 0.15 × 12.50
23. 6.5 × 0.7
© Houghton Mifflin Harcourt Publishing Company
Multiply. Write each product in simplest form.
Reading Start-Up
Visualize Vocabulary
Use the ✔ words to complete the graphic. You may put more
than one word in each box.
3
_
, 3:4
4
0.25
Reviewing Fractions and Decimals
Vocabulary
Review Words
✔ decimal (decimal)
✔ equivalent fractions
(fracciones equivalentes)
denominator
(denominador)
✔ fraction (fracción)
mixed number
(número mixto)
numerator (numerador)
✔ ratio (razón)
✔ simplest form (mínima
expresión)
Preview Words
2 _
_
= 69
3
4
_
→ _12
8
Understand Vocabulary
equivalent decimals
(decimales equivalentes)
model (modelo)
percent (porcentaje)
proportional reasoning
(razonamiento
proporcional)
© Houghton Mifflin Harcourt Publishing Company
Match the term on the left to the correct expression on the right.
1. percent
A. A ratio that compares a number to 100.
2. model
B. Decimals that name the same amount.
3. equivalent
decimals
C. Something that represents another thing.
Active Reading
Pyramid Before beginning the module, create
a pyramid to help you organize what you learn.
Label one side “Decimals,” one side “Fractions,” and
the other side “Percents.” As you study the module,
write important vocabulary and other notes on
the appropriate side.
Module 9
233
MODULE 9
Unpacking the TEKS
Understanding the TEKS and the vocabulary terms in the TEKS
will help you know exactly what you are expected to learn in this
module.
Generate equivalent forms
of fractions, decimals, and
percents using real-world
problems including problems
that involve money.
Key Vocabulary
equivalent expressions
(expresiones equivalentes)
Expressions that have the
same value.
What It Means to You
You will learn to write numbers in various forms, including
fractions, decimals, and percents.
UNPACKING EXAMPLE 6.4.G
Little brown bats flap their wings
about _34 as fast as pipistrelle bats do.
Write this fraction as a decimal and as
a percent.
3
_
= 3 ÷ 4 = 0.75
4
0.75 = 75%
6.5.B
Solve real-world problems to
find the whole given a part
and the percent, to find the
part given the whole and the
percent, and to find the percent
given the part and the whole,
including the use of concrete
and pictorial models.
Divide the numerator by the denominator.
Move the decimal point 2 places to
the right.
What It Means to You
You will solve problems involving percent.
UNPACKING EXAMPLE 6.5.B
About 67% of a person’s total
(100%) body weight is water.
If Cameron weighs 90 pounds,
about how much of his weight
is water?
Key Vocabulary
Percent (porcentaje)
A ratio comparing a
number to 100.
67% of 90
67
___
· 90
100
67 90
= ___ · __
100 1
= 60.3
About 60.3 pounds of Cameron’s weight is water.
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the
unpacked.
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234
Unit 3
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Ted Kinsman/
Science Source
6.4.G
LESSON
9.1 Understanding Percent
?
Proportionality—
6.4.E Represent percents
with concrete models [and]
fractions. Also 6.4.F
ESSENTIAL QUESTION
How can you write a ratio as a percent?
6.4.E
EXPLORE ACTIVITY 1
Using a Grid to Model Percents
A percent is a ratio that compares a number to 100. The symbol % is
used to show a percent.
17% is equivalent to
17
• ____
100
• 17 to 100
• 17:100
The free-throw ratios for three basketball players are shown.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Comstock/Getty
Images
33
15
17
Player 2: ___
Player 3: ___
Player 1: ___
25
50
20
A Rewrite each ratio as a number compared to 100. Then shade the
grid to represent the free-throw ratio.
17 = _____
Player 1: ___
25
100
33 = _____
Player 2: ___
50
100
15 = _____
Player 3: ___
20
100
B Which player has the greatest free-throw ratio?
How is this shown on the grids?
C Use a percent to describe each player’s free-throw ratio. Write the
percents in order from least to greatest.
D How did you determine how many squares to shade on each grid?
Lesson 9.1
235
EXPLORE ACTIVITY 2
6.4.E
Connecting Fractions and Percents
You can use a percent bar model to model a ratio expressed as a fraction
and to find an equivalent percent.
A Use a percent bar model to find an equivalent percent for _14 .
Draw a model to represent 100 and divide it into fourths. Shade _14 .
0
1
4
0%
1
100%
%
1
_
of 100 = 25, so _14 of 100% =
4
Tell which operation you can use to find _14 of 100.
Then find _14 of 100%.
B Use a percent bar model to find an equivalent percent for _13 .
Draw a model and divide it into thirds. Shade _13 .
0%
1
3
1
100%
%
1
_
of 100 = 33 _13, so _13 of 100% =
3
%
Tell which operation you can use to find _13 of 100.
Then find _13 of 100%.
Reflect
1.
236
Critique Reasoning Jo says she can find the percent equivalent of _34 by
multiplying the percent equivalent of _14 by 3. How can you use a percent
bar model to support this claim?
Unit 3
© Houghton Mifflin Harcourt Publishing Company
0
Using Benchmarks and
Proportional Reasoning
You can use certain benchmark percents to write other percents and
to estimate fractions.
Math On the Spot
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1
10
1
4
1
3
1
2
2
3
3
4
0
1
0
10%
25%
50%
1
33 3 %
75%
100
2
66 3 %
EXAMPL 1
EXAMPLE
6.4.F
3
A Find an equivalent percent for __
.
10
STEP 1
3
as a multiple of a benchmark fraction.
Write __
10
3
1
__
= 3 · __
10
10
STEP 2
1
Find an equivalent percent for __
.
10
1
__
= 10%
10
© Houghton Mifflin Harcourt Publishing Company
STEP 3
3
1 + ___
1
1
= ___
+ ___
Think: ___
10
10
10
10
Use the number lines to find the
1 .
equivalent percent for ___
10
Math Talk
Mathematical Processes
Explain how you could use
equivalent ratios to write
3
__
as a percent.
10
Multiply.
3
1
__
= 3 · __
= 3 · 10% = 30%
10
10
B 76% of the students at a middle school bring their own lunch.
About what fraction of the students bring their own lunch?
STEP 1
Note that 76% is close to the benchmark 75%.
STEP 2
Find a fraction equivalent for 75%:
75% = _34
About _34 of the students bring their own lunch.
Lesson 9.1
237
YOUR TURN
Personal
Math Trainer
Use a benchmark to find an equivalent percent for each fraction.
9
2. __
10
Online Assessment
and Intervention
3. _25
4. 64% of the animals at an animal shelter are dogs. About what fraction
of the animals at the shelter are dogs?
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Guided Practice
9
1. Shade the grid to represent the ratio __
. Then find a percent
25
equivalent to the given ratio. (Explore Activity 1)
9×
__________ = _____ =
25 ×
100
2. Use the percent bar model to find the missing percent. (Explore Activity 2)
1
5
0
0%
1
100%
%
6
3. __
10
1
Benchmark: _____
4. _24
5. _45
Benchmark:
_____
4
6. 41% of the students at an art college want to be graphic designers. About
what fraction of the students want to be graphic designers? (Example 1)
?
?
ESSENTIAL QUESTION CHECK-IN
7. How do you write a ratio as a percent?
238
Unit 3
Benchmark:
_____
5
© Houghton Mifflin Harcourt Publishing Company
Identify a benchmark you can use to find an equivalent percent for each
ratio. Then find the equivalent percent. (Example 1)
Name
Class
Date
9.1 Independent Practice
Personal
Math Trainer
6.4.E, 6.4.F
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Online
Assessment and
Intervention
Shade the grid to represent the ratio. Then find the missing number.
23 = _____
8. ___
50
100
11 = _____
9. ___
20
100
10. Mark wants to use a grid like the ones in Exercises 8 and 9 to model the
percent equivalent of the fraction _32. How many grid squares should he
shade? What percent would his model show?
11. The ratios of saves to the number of save opportunities are given for three
9 _
17
relief pitchers: __
, 4, __
. Write each ratio as a percent. Order the percents
10 5 20
from least to greatest.
Circle the greater quantity.
© Houghton Mifflin Harcourt Publishing Company
12. _13 of a box of Corn Krinkles
50% of a box of Corn Krinkles
13. 30% of your minutes are used up
_1 of your minutes are used up
4
14. Multiple Representations Explain how you could write 35% as the sum
of two benchmark percents or as a multiple of a percent.
15. Use the percent bar model to find the missing percent.
0
0%
1
8
1
%
100%
Lesson 9.1
239
a. What is the total number of songs Carl downloaded
last year?
Carl’s Downloads
Type of song
16. Multistep Carl buys songs and downloads them to his
computer. The bar graph shows the numbers of each type
of song he downloaded last year.
Country
Rock
Classical
World
0
b. What fraction of the songs were country? Find the fraction
for each type of song. Write each fraction in simplest form
and give its percent equivalent.
5
10
15
20 25
Number of songs
Work Area
FOCUS ON HIGHER ORDER THINKING
17. Critique Reasoning Marcus bought a booklet of tickets to use at the
amusement park. He used 50% of the tickets on rides, _13 of the tickets on
video games, and the rest of the tickets in the batting cage. Marcus says
he used 10% of the tickets in the batting cage. Do you agree? Explain.
Fraction
1
_
5
Percent
20%
2
_
5
3
_
5
4
_
5
5
_
5
6
_
5
a. Analyze Relationships What is true when the numerator and
denominator of the fraction are equal? What is true when the
numerator is greater than the denominator?
b. Justify Reasoning What is the percent equivalent of _32? Use a pattern
like the one in the table to support your answer.
240
Unit 3
© Houghton Mifflin Harcourt Publishing Company
18. Look for a Pattern Complete the table.
LESSON
9.2
?
Percents, Fractions,
and Decimals
Proportionality—
6.4.G Generate equivalent
forms of fractions, decimals,
and percents using real-world
problems. Also 6.5.C.
ESSENTIAL QUESTION
How can you write equivalent percents, fractions, and decimals?
Writing Percents as Decimals
and Fractions
You can write a percent as an equivalent fraction or as an equivalent decimal.
Equivalent percents, decimals, and fractions all represent equal parts of the
same whole.
EXAMPL 1
EXAMPLE
Write the percent as a fraction.
35
35% = ___
100
STEP 2
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6.4.G
Lorenzo spends 35% of his budget on rent for his apartment. Write this
percent as a fraction and as a decimal.
STEP 1
Math On the Spot
Percent means per 100.
Write the fraction in simplest form.
÷5
Math Talk
Mathematical Processes
Explain why both the
numerator and
denominator in Step 2
are divided by 5.
35
7
35
___
= ___
= __
100
100 20
© Houghton Mifflin Harcourt Publishing Company
STEP 3
÷5
Write the percent as a decimal.
Write the fraction equivalent of 35%.
35
___
35% = 100
35
.
Write the decimal equivalent of ____
100
= 0.35
7
So, 35% written as a fraction is __
and written as a decimal is 0.35.
20
YOUR TURN
Write each percent as a fraction and as a decimal.
1. 15%
2. 48%
3.
80%
4. 75%
5.
36%
6. 40%
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Online Assessment
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Lesson 9.2
241
6.5.C
EXPLORE ACTIVITY
Modeling Decimal, Fraction, and
Percent Equivalencies
Using models can help you understand how decimals, fractions,
and percents are related.
A Model 0.78 by shading a 10-by-10 grid.
0.78 = _____,
100
out of a hundred, or
%.
B Model 1.42 by shading 10-by-10 grids.
1.42 = _____ + _____ = _____ = 1 _____.
100
100
100
100
1.42 = 100% +
%=
%
C Model 125% by shading 10-by-10 grids.
The model shows 100% +
125% = the decimal
% = 125%.
.
Reflect
7. Multiple Representations What decimal, fraction, and percent
equivalencies are shown in each model? Explain.
a.
b.
242
Unit 3
© Houghton Mifflin Harcourt Publishing Company
125% = ______ + ______ = ______ = 1 ______ = 1 _____.
100
100
100
100
Writing Fractions as Decimals
and Percents
You can write some fractions as percents by writing an equivalent fraction
with a denominator of 100. This method is useful when the fraction has a
denominator that is a factor or a multiple of 100. If a fraction does not have a
denominator that is a factor or multiple of 100, you can use long division.
EXAMPL 2
EXAMPLE
Write an equivalent fraction with a denominator of 100.
96
48
___
= ___
200
100
STEP 2
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6.4.G
96
A 96 out of 200 animals treated by a veterinarian are horses. Write ___
as a
200
decimal and as a percent.
STEP 1
Math On the Spot
Notice that the
denominator is a
multiple of 100.
Divide both the numerator and denominator by 2.
Write the decimal equivalent.
48
___
= 0.48
100
STEP 3
Write the percent equivalent.
48
___
= 48%
100
Percent means per 100.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Digital Vision/
Getty Images
Notice that the denominator is
not a factor or multiple of 100.
B _18 of the animals treated by the veterinarian are dogs. Write _18 as a decimal
and as a percent.
STEP 1
Use long division to divide the numerator by the denominator.
0.125
⎯
1
_
= 8⟌1.000
8
Add a decimal point and zeros to the right
of the numerator as needed.
-8
20
- 16
40
- 40
0
The decimal equivalent of _18 is 0.125.
STEP 2
Write the decimal as a percent.
125
0.125 = ____
1,000
Write the fraction equivalent of the decimal.
÷ 10
125
12.5
____
= ____
1,000
100
Write an equivalent fraction with
a denominator of 100.
÷ 10
12.5
____
= 12.5%
100
Write as a percent.
The percent equivalent of _18 is 12.5%.
Lesson 9.2
243
YOUR TURN
Personal
Math Trainer
Online Assessment
and Intervention
Write each fraction as a decimal and as a percent.
9
8. __
25
9. _78
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Guided Practice
1. Helene spends 12% of her budget on transportation expenses. Write this
percent as a fraction and as a decimal. (Example 1)
Model the decimal. Then write percent and fraction equivalents.
(Explore Activity)
2. 0.53
3. 1.07
7
4. __
of the packages
20
?
?
5. _38 of a pie
ESSENTIAL QUESTION CHECK-IN
6. How does the definition of percent help you write fraction and decimal
equivalents?
244
Unit 3
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Write each fraction as a decimal and as a percent. (Example 2)
Name
Class
Date
9.2 Independent Practice
Personal
Math Trainer
6.4.G, 6.5.C
my.hrw.com
Online
Assessment and
Intervention
Write each percent as a fraction and as a decimal.
7. 72% full
10. 5% tax
8. 25% successes
11. 37% profit
9. 500% increase
12. 165% improvement
Write each fraction as a decimal and as a percent.
13. _58 of an inch
258
14. ___
of the contestants
300
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16. The poster shows how many of its games
the football team has won so far. Express
this information as a fraction, a percent,
and as a decimal.
350
15. ___
of the revenue
100
17. Justine answered 68 questions correctly on
an 80-question test. Express this amount as
a fraction, percent, and decimal.
Each diagram is made of smaller, identical pieces. Tell how many pieces
you would shade to model the given percent.
18. 75%
19. 25%
Lesson 9.2
245
20. Multiple Representations At Brian’s Bookstore, 0.3 of the shelves hold
7
mysteries, 25% of the shelves hold travel books, and __
of the shelves hold
20
children’s books. Which type of book covers the most shelf space in the
store? Explain how you arrived at your answer.
Work Area
FOCUS ON HIGHER ORDER THINKING
21. Critical Thinking A newspaper article reports the results of an election
between two candidates. The article says that Smith received 60% of
the votes and that Murphy received _13 of the votes. A reader writes in to
complain that the article cannot be accurate. What reason might the
reader have to say this?
22. Represent Real-World Problems Evan budgets $2,000 a month to
spend on living expenses for his family. Complete the table to express the
portion spent on each cost as a percent, fraction, and decimal.
Food: $500
Rent: $1,200
Transportation: $300
Fraction
Percent
23. Communicate Mathematical Ideas Find the sum of each row in the
table. Explain why these sums make sense.
24. Explain the Error Your friend says that 14.5% is equivalent to the
decimal 14.5. Explain why your friend is incorrect by comparing the
fractional equivalents of 14.5% and 14.5.
246
Unit 3
© Houghton Mifflin Harcourt Publishing Company
Decimal
LESSON
9.3
?
Solving Percent
Problems
Proportionality—
6.5.B Solve real-world
problems involving percent.
Also 6.4.G
ESSENTIAL QUESTION
How do you use percents to solve problems?
6.5.B
EXPLORE ACTIVITY
Modeling a Percent Problem
You can use a model to solve a percent problem.
A sports store received a shipment of 400 baseball gloves. 30% were
left-handed. How many left-handed gloves were in the shipment?
A Use the diagram to solve this problem.
30% means 30 out of
.
There were
left-handed gloves
for every 100 baseball gloves.
30
100
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Photodisc/
Getty Images
Complete the diagram to model this situation.
400
B Describe how the diagram models the shipment of gloves.
C Explain how you can use the diagram to find the total number of
left-handed gloves in the shipment.
D Use a bar model to solve this problem. The bar represents 100%, or the
entire shipment of 400 gloves. The bar is divided into 10 equal parts.
Complete the labels along the bottom of the bar.
0%
10%
20%
0
40
80
30%
40%
50%
60%
70%
80%
90% 100%
400
Lesson 9.3
247
EXPLORE ACTIVITY (cont’d)
Reflect
1.
Justify Reasoning How did you determine the labels along the
bottom of the bar model in Step D?
2.
Communicate Mathematical Ideas How can you use the bar model
to find the number of left-handed gloves?
Finding a Percent of a Number
my.hrw.com
Proportional Reasoning
×4
30
? ←part
___
= ___
100
400 ←whole
Multiplication
30
of 400
30% of 400 = ___
100
30
= ___
× 400
100
×4
= 120
120
= ___
400
EXAMPLE 1
6.5.B
A Use proportional reasoning to find 28% of 25.
Math Talk
STEP 1
Mathematical Processes
Could you also use the
28
?
proportion ___
= __
to find
100
25
28% of 25? Explain.
Write a proportion comparing the percent to the ratio
of part to whole.
?
28
__
= ___
25
100
STEP 2
Notice that 25 is a factor of 100.
Find the multiplication factor.
×4
part → __
?
28
___
whole → 25 = 100
Since 25 · 4 = 100, find what
number times 4 equals 28.
×4
STEP 3
Find the numerator.
28
7
__
= ___
25
100
28% of 25 is 7.
248
Unit 3
Since 4 · 7 = 28, 28% of 25 = 7.
© Houghton Mifflin Harcourt Publishing Company
Math On the Spot
A percent is equivalent to the ratio of a part to a whole. To find a percent of a
number, you can write a ratio to represent the percent, and find an equivalent
ratio that compares the part to the whole.
The word of indicates
multiplication.
To find 30% of 400, you can use:
B Multiply by a fraction to find 35% of 60.
STEP 1
Write the percent as a fraction.
35
35% of 60 = ___
of 60
100
STEP 2
Multiply.
35
35
___
of 60 = ___
· 60
100
100
2,100
= ____
100
= 21
Animated
Math
Simplify.
35% of 60 is 21.
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C Multiply by a decimal to find 5% of 180.
STEP 1
Write the percent as a decimal.
5
5% = ___
= 0.05
100
STEP 2
Multiply.
180 · 0.05 = 9
5% of 180 is 9
Reflect
Analyze Relationships In B, the percent is 35%. What is the part and
what is the whole?
4.
Communicate Mathematical Ideas Explain how to use proportional
reasoning to find 35% of 600.
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3.
YOUR TURN
Personal
Math Trainer
Find the percent of each number.
5.
38% of 50
6.
27% of 300
7.
60% of 75
Online Assessment
and Intervention
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Lesson 9.3
249
Find a Percent Given a Part
and a Whole
Math On the Spot
You can use proportional reasoning to solve problems in which you need to
find a percent.
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EXAMPLE 2
6.5.B
The school principal spent $2,000 to buy some new computer equipment.
Of this money, $120 was used to buy some new keyboards. What percent
of the money was spent on keyboards?
STEP 1
Since you want to know the part of the money spent on
keyboards, compare the part to the whole.
part → _____
$120
whole → $2,000
STEP 2
Write a proportion comparing the percent to the ratio of part to
whole.
part → ___
120 ← part
?
____
whole → 100 = 2,000 ← whole
STEP 3
Find the multiplication factor.
×20
?
120
___
= ____
100
2,000
×20
STEP 4
Since 100 · 20 = 2,000, find what number
times 20 equals 120.
Find the numerator.
6
120
___
= ____
100
2,000
Since 20 · 6 = 120, the percent is 6%.
Reflect
8.
Communicate Mathematical Ideas Write 57% as a ratio. Which
number in the ratio represents the part and which number represents
the whole? Explain.
YOUR TURN
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Math Trainer
Online Assessment
and Intervention
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250
Unit 3
9.
Out of the 25 students in Mrs. Green’s class, 19 have a pet. What percent
of the students in Mrs. Green’s class have a pet?
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The principal spent 6% of the money on keyboards.
Finding a Whole Given a Part
and a Percent
You can use proportional reasoning to solve problems in which you know
a part and a percent and need to find the whole.
Math On the Spot
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EXAMPL 3
EXAMPLE
6.5.B
Twelve of the students in the school choir like to sing solos. These 12
students make up 24% of the choir. How many students are in the choir?
Method 1: Use a concrete model.
24% represents 12 students.
100 squares represent 100%.
24 squares represent 24%.
12 _
Since __
= 12, 1 square represents _12 student.
24
100 · _12 = 50, so 100 squares represent
50 students.
Method 2: Use a proportion.
part → __
24 ← part
12 ___
whole → ? = 100 ← whole
×2
12 ___
24
__
= 100
?
×2
12 ___
24
__
= 100
50
Math Talk
Write a proportion.12 students
represent 24%.
Mathematical Processes
Suppose 10 more students
join the choir. None of
them are soloists. What
percent are soloists
now?
Since 12 · 2 = 24, find what number times
2 = 100.
Since 50 · 2 = 100, the denominator is 50.
© Houghton Mifflin Harcourt Publishing Company
There are 50 students in the choir.
Reflect
10.
Multiple Representations Sixteen students in the school band play
clarinet. Clarinet players make up 20% of the band. Use a bar model to
find the number of students in the school band.
0%
10%
0
20%
30%
40%
50%
60%
70%
80%
90% 100%
16
Personal
Math Trainer
YOUR TURN
11. 6 is 30% of ______.
12. 15% of _____ is 75.
Online Assessment
and Intervention
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Lesson 9.3
251
Guided Practice
1. A store has 300 televisions on order, and 80% are high definition. (Explore
Activity)
a. Use the bar model and complete the bottom of the bar.
0%
10%
0
30
20%
30%
40%
50%
60%
70%
80%
90% 100%
300
b. Complete the diagram to model this situation.
80
100
300
c. How many televisions on the order are high definition?
2. Use proportional reasoning to find 65%
of 200. (Example 1)
part
→
?
_____ = _____
whole → 100
← part
3. Use multiplication to find 5% of 180.
(Example 1)
5 of 180 = ____
5
____
100
180
100
← whole
= _____ =
100
.
4. Alana spent $21 of her $300 paycheck on
a gift. What percent of her paycheck was
spent on the gift? (Example 2)
Alana spent
the gift.
?
?
5% of 180 is
5. At Pizza Pi, 9% of the pizzas made last
week had extra cheese. If 27 pizzas had
extra cheese, how many pizzas in all were
made last week? (Example 3)
of her paycheck on
There were
ESSENTIAL QUESTION CHECK-IN
6. How can you use proportional reasoning to solve problems involving
percent.
252
Unit 3
.
pizzas made last week.
© Houghton Mifflin Harcourt Publishing Company
65% of 200 is
Name
Class
Date
9.3 Independent Practice
Personal
Math Trainer
6.5.B
my.hrw.com
Online
Assessment and
Intervention
Find the percent of each number.
7. 64% of 75 tiles
8. 20% of 70 plants
10. 85% of 40 e-mails
9. 32% of 25 pages
11. 72% of 350 friends
12. 5% of 220 files
Complete each sentence.
13. 4 students is
15.
% of 20 students.
% of 50 shirts is 35 shirts.
16.
% of 25 doctors.
% of 200 miles is 150 miles.
17. 4% of
days is 56 days.
18. 60 minutes is 20% of
19. 80% of
games is 32 games.
20. 360 kilometers is 24% of
21. 75% of
peaches is 15 peaches.
22. 9 stores is 3% of
23. At a shelter, 15% of the dogs are puppies.
There are 60 dogs at the shelter.
How many are puppies?
minutes.
kilometers.
stores.
24. Carl has 200 songs on his MP3 player. Of these
songs, 24 are country songs. What percent of
puppies
25. Consumer Math The sales tax in the town
where Amanda lives is 7%. Amanda paid
$35 in sales tax on a new stereo. What was
© Houghton Mifflin Harcourt Publishing Company
14. 2 doctors is
Carl’s songs are country songs?
26. Financial Literacy Ashton is saving money to
buy a new bike. He needs $120 but has only
saved 60% so far. How much more money
the price of the stereo?
does Ashton need to buy the scooter?
27. Consumer Math Monica paid sales tax of $1.50 when she bought a new
bike helmet. If the sales tax rate was 5%, how much did the store charge
for the helmet before tax?
28. Use the circle graph to determine how many hours per day Becky spends
on each activity.
Becky’s Day
School:
hours
Eating:
hours
Sleep:
Homework:
Free time:
hours
hours
Eating
10%
Free time
15%
Homework
10%
Sleep
40%
School
25%
hours
Lesson 9.3
253
FOCUS ON HIGHER ORDER THINKING
Work Area
29. Multistep Marc ordered a rug. He gave a deposit of 30% of the cost and
will pay the rest when the rug is delivered. If the deposit was $75, how
much more does Marc owe? Explain how you found your answer.
30. Earth Science Your weight on different planets is affected by gravity. An
object that weighs 150 pounds on Earth weighs only 56.55 pounds on
Mars. The same object weighs only 24.9 pounds on the Moon.
a. What percent of an object’s Earth weight is its weight on Mars and on
the Moon?
b. Suppose x represents an object’s weight on Earth. Write two expressions:
one that you can use to find the object’s weight on Mars and another
that you can use to write the object’s weight on the Moon.
c. The space suit Neil Armstrong wore when he stepped on the Moon
for the first time weighed about 180 pounds on Earth. How much did
it weigh on the Moon?
31. Explain the Error Fifteen students in the band play
clarinet. These 15 students make up 12% of the band.
?
12
Your friend used the proportion ___
= __
15 to find the
100
number of students in the band. Explain why your
friend is incorrect and use the grid to find the correct
answer.
254
Unit 3
© Houghton Mifflin Harcourt Publishing Company
d. What If? If you could travel to Jupiter, your weight would be 236.4%
of your Earth weight. How much would Neil Armstrong’s space suit
weigh on Jupiter?
MODULE QUIZ
Ready
Personal
Math Trainer
9.1 Understanding Percent
Online Assessment
and Intervention
my.hrw.com
Shade the grid and write the equivalent percent for each fraction.
19
1. __
50
13
2. __
20
9.2 Percents, Fractions, and Decimals
Write each number in two equivalent forms.
3. _53
4. 62.5%
5. 0.24
31
6. __
50
7
7. Selma spent __
10 of her allowance on a new backpack.
What percent of her allowance did she spend?
9.3 Solving Percent Problems
Complete each sentence.
8. 12 is 30% of
© Houghton Mifflin Harcourt Publishing Company
10. 18 is
.
9. 45% of 20 is
.
% of 30.
11. 56 is 80% of
.
12. A pack of cinnamon-scented pencils sells for $4.00. What
is the sales tax rate if the total cost of the pencils is $4.32?
ESSENTIAL QUESTION
13. How can you solve problems involving percents?
Module 9
255
Personal
Math Trainer
MODULE 9 MIXED REVIEW
Texas Test Prep
Selected Response
1. What percent does this shaded grid
represent?
52%
D 58%
2. Which expression is not equal to one
fourth of 52?
A 0.25 • 52
B 4% of 52
52 ÷ 4
52
D __
4
3. Approximately _45 of U.S. homeowners have
a cell phone. What percent of homeowners
do not have a cell phone?
A 20%
B 45%
C
A $96
C
B $108
D $180
55%
A 10
C
B 160
D 900
4. The ratio of rock music to total CDs that
25
Ella owns is __
40. Paolo has 50 rock music
CDs. The ratio of rock music to total CDs in
his collection is equivalent to the ratio of
rock music to total CDs in Ella’s collection.
How many CDs do they own?
200
7. Dominic answered 43 of the 50 questions
on his spelling test correctly. Which
decimal represents the fraction of
problems he answered incorrectly?
A 0.07
C
B 0.14
D 0.93
0.86
Gridded Response
8. Jen bought some bagels. The ratio of the
number of sesame bagels to the number
of plain bagels that she bought is 1:3. Find
the decimal equivalent of the percent of
the bagels that are plain.
D 80%
256
$162
.
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
A 65
C
5
5
5
5
5
5
120
6
6
6
6
6
6
B 80
D 130
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
Unit 3
© Houghton Mifflin Harcourt Publishing Company
B 48%
C
5. Gabriel saves 40% of his monthly paycheck
for college. He earned $270 last month.
How much money did Gabriel save for
college?
6. Forty children from an after-school club
went to the matinee. This is 25% of the
children in the club. How many children
are in the club?
A 42%
C
my.hrw.com
Online
Assessment and
Intervention
Review
UNIT 3
Study Guide
MODULE
?
7
Representing Ratios
and Rates
Key Vocabulary
ESSENTIAL QUESTION
How can you use ratios and rates to solve real-world problems?
equivalent ratios (razones
equivalentes)
rate (tasa)
ratio (razón)
unit rate (tasa unitaria)
EXAMPLE 1
Tina pays $45.50 for 13 boxes of wheat crackers. What is the
unit price?
$45.50
$3.50
_______
= _____
13 boxes 1 box
The unit price is $3.50 per box of crackers.
EXAMPLE 2
A trail mix recipe calls for 3 cups of raisins and 4 cups of peanuts.
Mitt made trail mix for a party and used 5 cups of raisins and 6 cups
of peanuts. Did Mitt use the correct ratio of raisins to peanuts?
3 cups of raisins
_____________
4 cups of peanuts
3
The ratio of raisins to peanuts in the recipe is __
.
4
5 cups of raisins
_____________
6 cups of peanuts
Mitt used a ratio of _56_.
3 _
9
_
× 3 = __
4 3 12
5 _
10
_
× 2 = __
6 2 12
10
9
__
< __
12 12
Mitt used a higher ratio of raisins to peanuts in his trail mix.
© Houghton Mifflin Harcourt Publishing Company
EXERCISES
Write three equivalent ratios for each ratio. (Lesson 7.1)
1.
18
__
6
5
2. __
45
3. _35
4.
To make a dark orange color, Ron mixes 3 ounces of red paint with
2 ounces of yellow paint. Write the ratio of red paint to yellow paint
three ways. (Lesson 7.1)
5.
A box of a dozen fruit tarts costs $15.00. What is the cost of one fruit
tart? (Lesson 7.2)
Compare the ratios. (Lesson 7.3)
6.
_2
5
_3
4
7. _92
10
__
7
2
8. __
11
3
__
12
9. _67
_8
9
Unit 3
257
MODULE
?
8
Applying Ratios and Rates
Key Vocabulary
conversion factor (factor
de conversión)
proportion (proporción)
scale drawing (dibujo a
escala)
scale factor (factor de
escala)
ESSENTIAL QUESTION
How can you use ratios and rates to solve real-world problems?
EXAMPLE 1
Jessica earns $5 for each dog she walks. Complete the
table, describe the rule, and tell whether the relationship
is additive or multiplicative. Then graph the ordered pairs
on a coordinate plane.
1
2
3
4
5
Profit ($)
5
10
15
20
25
Jessica’s profit is the number of dogs walked multiplied
by $5. The relationship is multiplicative.
(5, 25)
25
Profit ($)
Number of dogs
y
(4, 20)
20
(3, 15)
15
(2, 10)
10
(1, 5)
5
O
EXAMPLE 2
Kim’s softball team drank 3 gallons of water during practice.
How many cups of water did the team drink?
16 cups
? cups
______
= _______
1 gallon 3 gallons
16 × 3 __
_____
= 48
1×3
3
1
2
x
3
4
5
Number of dogs
16 cups
48 cups
______
= _______
1 gallon 3 gallons
The team drank 48 cups of water.
EXERCISES
4
Total savings
9
6
8
10
20
16
12
8
4
O
2. There are 2 hydrogen atoms and 1 oxygen atom in a water
molecule. Complete the table, and list the equivalent ratios
shown on the table. (Lesson 8.1, 8.2)
Hydrogen atoms
Oxygen atoms
8
16
20
6
3. Sam can solve 30 multiplication problems in 2 minutes. How many
can he solve in 20 minutes? (Lesson 8.3)
258
Unit 3
x
2
4
6
8 10
New savings
© Houghton Mifflin Harcourt Publishing Company
New savings
y
Total savings
1. Thaddeus already has $5 saved. He wants to save more to
buy a book. Complete the table, and graph the ordered
pairs on the coordinate graph. (Lesson 8.1, 8.2)
4. A male Chihuahua weighs 5 pounds. How many ounces does he
weigh? (Lesson 8.4)
MODULE
?
9
Percents
ESSENTIAL QUESTION
How can you use percents to solve real-world problems?
EXAMPLE 1
7
Find an equivalent percent for __
10 .
1
10
1
4
1
3
1
2
Find an equivalent percent for _15 .
2
3
3
4
0
0
0
1
_
5
1
1
10%
25%
50%
75%
1
33 3 %
7
1
__
= 7 · __
10
10
100
2
66 3 %
7
__
= 7 · 10%
10
7
__
= 70%
10
0%
%
100%
1
_
of 100 = 20, so _15 of 100% = 20%
5
1
_
= 20%
5
EXAMPLE 2
Thirteen of the 50 states in the United States do not touch the
13
ocean. Write __
50 as a decimal and a percent.
© Houghton Mifflin Harcourt Publishing Company
13 ___
__
= 26
50 100
26
___
= 0.26
100
0.26 = 26%
13
__
= 0.26 = 26%
50
EXAMPLE 3
Buckner put $60 of his $400 paycheck into his savings account. Find
the percent of his paycheck that Buckner saved.
60
?
___
= ___
400 100
60 ÷ 4
15
______
= ___
400 ÷ 4 100
Buckner saved 15% of his paycheck.
EXERCISES
Write each fraction as a decimal and a percent. (Lessons 9.1, 9.2)
1. _34
3. _85
7
2. __
20
Complete each statement. (Lessons 9.1, 9.2)
4. 25% of 200 is
.
5. 16 is
of 20.
6. 21 is 70% of
.
Unit 3
259
7. 42 of the 150 employees at Carlo’s Car Repair wear
contact lenses. What percent of the employees wear
contact lenses? (Lesson 9.3)
8. Last week at Best Bargain, 75% of the computers sold
were laptops. If 340 computers were sold last week,
how many were laptops? (Lesson 9.3)
Unit 3 Performance Tasks
1.
CAREERS IN MATH
Residential Builder Kaylee, a residential builder,
is working on a paint budget for a custom-designed home she is building.
A gallon of paint costs $38.50, and its label says it covers about 350 square feet.
a. Explain how to calculate the cost of paint per square foot. Find this
value. Show your work.
b. Kaylee measured the room she wants to paint and calculated a total
area of 825 square feet. If the paint is only available in one-gallon
cans, how many cans of paint should she buy? Justify your answer.
2. Davette wants to buy flannel sheets. She reads that a weight of at least
190 grams per square meter is considered high quality.
b. Davette finds 3 more options for flannel sheets:
Option 1: 1,100 g of flannel in 6 square meters, $45
Option 2: 1,260 g of flannel in 6.6 square meters, $42
Option 3: 1,300 g of flannel in 6.5 square meters, $52
She would like to buy the sheet that meets her requirements for
high quality and has the lowest price per square meter. Which
option should she buy? Justify your answer.
260
Unit 3
© Houghton Mifflin Harcourt Publishing Company
a. Davette finds a sheet that has a weight of 920 grams for 5 square
meters. Does this sheet satisfy the requirement for high-quality
sheets? If not, what should the weight be for 5 square meters? Explain.
Personal
Math Trainer
UNIT 3 MIXED REVIEW
Texas Test Prep
1. The deepest part of a swimming pool is
12 feet deep. The shallowest part of the
pool is 3 feet deep. What is the ratio of the
depth of the deepest part of the pool to the
depth of the shallowest part of the pool?
5. The graph below represents Donovan’s
speed while riding his bike.
10
Distance (km)
Selected Response
my.hrw.com
Online
Assessment and
Intervention
A 4:1
B 12:15
8
6
4
2
O
C 1:4
2
4
6
8 10
Time (min)
D 15:12
2. How many centimeters are in 15 meters?
Which would be an ordered pair on
the line?
A 0.15 centimeters
A (1, 3)
B 1.5 centimeters
B (2, 2)
C 150 centimeters
C (6, 4)
D 1,500 centimeters
D (9, 3)
3. Barbara can walk 3,200 meters in
24 minutes. How far can she walk in
3 minutes?
Hot !
Tip
A 320 meters
Read the graph or diagram as
closely as you read the actual
test question. These visual aids
contain important information.
6. Which percent does this shaded grid
represent?
B 400 meters
C 640 meters
© Houghton Mifflin Harcourt Publishing Company
D 720 meters
4. The table below shows the number
of windows and panes of glass in the
windows.
Windows
2
3
4
5
Panes
12
18
24
30
A 42%
B 48%
Which represents the number of panes?
A windows × 5
C 52%
D 58%
B windows × 6
C windows + 10
D windows + 15
Unit 3
261
7. Ivan saves 20% of his monthly paycheck
for music equipment. He earned $335 last
month. How much money did Ivan save for
music equipment?
11. A recipe calls for 6 cups of water and 4 cups
of flour. If the recipe is increased, how many
cups of water should be used with 6 cups
of flour?
A $65
.
B $67
0
0
0
0
0
0
C $70
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
D $75
8. How many 0.6-liter glasses can you fill up
with a 4.5-liter pitcher?
A 1.33 glasses
7
7
7
7
7
7
B 3.9 glasses
8
8
8
8
8
8
C 7.3 glasses
9
9
9
9
9
9
D 7.5 glasses
9. Which shows the integers in order from
greatest to least?
A 22, 8, 7, 2, -11
B 2, 7, 8, -11, 22
D 22, -11, 8, 7, 2
10. Melinda bought 6 bowls for $13.20. What
was the unit rate, in dollars?
.
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
0
0
0
0
0
0
5
5
5
5
5
5
1
1
1
1
1
1
6
6
6
6
6
6
2
2
2
2
2
2
7
7
7
7
7
7
3
3
3
3
3
3
8
8
8
8
8
8
4
4
4
4
4
4
9
9
9
9
9
9
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
© Houghton Mifflin Harcourt Publishing Company
.
Gridded Response
Unit 3
Estimate your answer before
solving the question. Use
your estimate to check the
reasonableness of your answer.
12. Broderick answered 21 of the 25 questions
on his history test correctly. What decimal
represents the fraction of problems he
answered incorrectly?
C -11, 2, 7, 8, 22
262
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